Suggested Readings
Most of my research focuses on category theory and categorical logic. This webpage gathers a bit of material and is designed for bachelor and master students that want to have a first encounter with these topics.
The biggest advantage of this webpage with respect to a much more ambitious project like the nlab is its incompleteness. Especially for a very young student a lot of information is very often the same as too much information. In this sense, the point of view that I try to offer is both a bias and an opportunity.
The sections below are fairly independent with the exception of the one on Categorical Logic, which should not be browsed without a basic knowledge of Category Theory. The meaning of the word basic is clarified at the beginning of the CT section.
Sometimes links might be broken, please (do not?) contact me if they stop working. This page will never find its final version, and will be a never ending work in progress. If you like any of these topics, and you want to get in touch with me, just drop me a line!
Category Theory
There exists a plethora of introductions to category theory. Some of them are very famous both for the clarity of their exposition and for their historical importance. Among those one should definitely list the evergreen Categories for the working mathematician by Saunders MacLane.
Yet I do not consider that book to be the best introduction to the subject for a bachelor student. Instead, the following two references cover the basics of category theory, offering incredible clarity and an enormous amount of examples.
I suggest these two books with the same motto that characterize this webpage, they go directly to the point, covering only the essential and giving the student a good foundation on which to build future knowledge.

by Leinster. A very good introduction to the subject, especially for bachelor students.

by Riehl. A bit more complete than the previous one, it also includes Kan extensions and monads.
Once the basic material above is well understood, one can enter in the zoo of category theory at work and meet its most different contextualizations. The following two books offer a very classical perspective on several incarnations of category theory.

Categorical Foundations.
edited by Pedicchio and Tholen. Categorical approaches to general topology, algebra and universal algebra. A concise introduction to Sheaf theory.

Locally presentable and accessible categories.
by Adamek and Rosický. Chapters 14. In addition to the topics in the title, it is a very sound introduction also for orthogonality, injectivity and algebraic theories. If you are looking for the shortest path to learn the core of the small object argument, click here.
From the point of view of the most recent trends in category theory, both the two books will seem a bit old fashioned, and indeed they are. This does not make them less interesting, or relevant. Also, starting from the most recent trends, ignoring completely the more established ones, does not come without risks.
Notice that I am neglecting on purpose Homotopy theory because it is a very special flavor of category theory that would not fit with the ambition of generality of this list nor with the special taste of his author. That is not the case of locally presentable and accessible categories: in a sense, those offer a theory of tame categories and play a relevant role in many fields, ranging from homotopy theory (combinatorial model categories) to abstract model theory (abstract elementary classes).
In my personal experience, Category theory has proven to be a seductive and dangerous field, especially for youngsters. Please avoid the Scuttle paradox.
Formal Category Theory
We are all indebted with the Australian school of category theory for the invention of formal category theory. By this term I intend anything related to the topic of enriched categories, 2categories and bicategories. Unfortunately it is very hard to find good and complete texts on this topic. Luckily, since this page is dedicated to youngsters, I have no ambition of providing a complete reference, and I will try to stay on the basics.

Handbook of categorical algebra I.
by Borceux. Chapters 7 and 8: Bicategories, Distributors and Internal categories.

Handbook of categorical algebra II.
by Borceux. Chapters 6 and 8: Enriched categories, Fibred categories.

by Myers.

by Benabou.

Kan Extensions in Enriched Category Theory.
by Dubuc.

by Lack.

Elementary observations on 2categorical limits.
by Kelly.
The Australian tradition has various applications, the one that is closest to my heart is the intense study of the properties of the 2category Cat of locally small categories and functors.
In this fashion, my favorite topic in formal category theory is that of Yoneda structures, that play a relevant role in an abstract treatment of semantic representation of logic. It is almost impossible to provide a concise reference for this topic, I can recommend to start from the following link.

An Exegesis of Yoneda Structures.
by Alexander Campbell.
Categorical Logic
Depending on their background, people tend to identify categorical logic with some of its parts. People from computer science think that categorical logic is a framework that offers a semantic for type theory and λcalculus, pure mathematicians are mostly aware of topos theory, because of its connection with the rest of mathematics, and finally just not many people are interested in the categorical approach to universal algebra and model theory.
This sad fragmentation of the field is encouraged by the available literature. Most of the books on categorical logic concentrate just on one of the aspects abovementioned, and a complete source does not exist to my knowledge. While we wait for such a bible to appear, I suggest everyone to start the journey trough categorical logic via the historical book below.

The history of categorical logic 1963–1977.
by Marquis and Reyes.
While the book leaves uncovered almost 50 years of beautiful mathematics, it shows quite strongly that categorical logic is a unite field, where the subjects above melt into a unique discourse. Please, read it slowly and carefully: history is important and we should guard it. Once one has some understanding of what categorical logic is about, these are the books that I suggest to start with.
Universal algebra

The category theoretic understanding of Universal Algebra: Lawvere Theories and Monads.
By Hyland and Power.

Algebraic Theories: a categorical introduction to General Algebra.
By Vitale, Rosický and Adámek.

Enriched algebraic theories and monads for a system of arities.
By LucyshynWright.

Monads of sets
By Manes in Handbook of Algebra.
Theoretical Computer Science

Category theory for computing science.
By Barr and Wells.

Introduction to higherorder Categorical Logic.
By Lambek and Scott.

Categorical logic and type theory.
By Jacobs.

Algebra, categories and databases.
By Plotkin in Handbook of Algebra.

Some aspects of categories in computer science.
By Scott in Handbook of Algebra.
Topos theory
Topos theory is the elephant in the room of categorical logic. An elephant transformed into a Behemoth from the mithological narrative that sorrounds the most technical aspects of the topic. Despite its oversensationalisation, topos theory is really, just, absolutely, beautiful. I only reccomend not indulging too much in juvenile romanticisms, and instead, sitting down to study it like any other topic — with patience and devotion. Do not expect to absorb the topic in less than a full year of study. For geometers, give a chance to its logical aspects. For logicians, listen to the story that geometers have to tell. The best topos theorists are by far those who manage to have both points of view.
It is not easy to decide where to start one's journey into topos theory. My reccomendation will be the following. Start from the two souces below, and if you are very fluent with category theory, do check out the third. Then (and only then) start reading the Elephant, untill it sticks.

An informal introduction to topos theory.
By Leinster.

Sheaves in Geometry and Logic.
By Moerdijk and MacLane.

Topos Theory.
By Johnstone.

Sketches of an elephant.
By Johnstone.
Categorical Model Theory

Accessible categories: the foundations of categorical model theory.
by Paré and Makkai.

Locally presentable and accessible categories.
by Adamek and Rosicky. Chapters 3 and 5.
Selected papers
A lot of very good mathematics does not lie in a book yet, this very short subsection lists some very good papers.

Concrete categories and infinitary languages.
by Rosicky.

Concrete dualities.
by Tholen and Porst.

Stone duality for first order logic.
by Makkai.

FirstOrder Logical Duality.
by Awodey and Forssell.

A Note on Russell's Paradox in locally cartesian closed categories.
by Pitts and Taylor.

Metric spaces, generalized logic, closed categories.
by Lawvere. Read it twice, I am still learning from this article. Don't underestimate the importance of Sec. 3.

Diagonal Arguments and Cartesian Closed Categories.
by Lawvere.

Class forcing and topos theory.
by Roberts.

Comparing material and structural set theories.
by Shulman.
Logic and foundational aspects of Type Theory
(Mathematical) Logic
Defining (mathematical) logic is too ambitious and very far for the general aim of this webpage. Students are generally introduced to the fragment of mathematical logic that is mostly studied in their department. As already stated above, this is both a limitation and an opportunity. Yet a mature researcher, or a student that wants to become a mature researcher, should be able to navigate the map of logic, being aware of its countries and their leaders. In order to restrain (which is very different from avoid) the consequences of the no true Scotsman fallacy, a first approach to this understanding should always be historical. For this purposes I suggest to give a look to the book series below.

Handbook of the History of Logic.
Vol. 311. The Rise of Modern Logic: from Leibniz to Frege. British Logic in the Nineteenth Century. Logic from Russell to Church. Sets and Extensions in the Twentieth Century. Logic and the Modalities in the Twentieth Century. The Many Valued and Nonmonotonic Turn in Logic. Inductive Logic. Logic: A History of its Central Concepts.
It is absolutely impossible to read entirely such an encyclopedia, its value is to relativize the areas of classical logic and to introduce the reader to some topics. Once one has some vague idea of the historical development of logic, it's time to study. The book below is a soft but extremely comprehensive introduction to many fields and subfields of logic.

Open book. A collection of teaching materials on mathematical logic aimed at a nonmathematical audience, intended for use in advanced logic courses as taught in many philosophy departments
Finally, let me focus on those fields that are better known to myself, namely: Model theory and Set theory.

Lectures in Logic and Set Theory. Volume 1: Mathematical Logic.
by Tourlakis.

Lectures in Logic and Set Theory. Volume 2: Set Theory.
by Tourlakis.

Appunti sul calcolo dei predicati.
by Berarducci. Italian. This is the shrinked italian analog of the first part of Vol. 1 by Tourlakis.

Calcolabilità e Teoremi di Gödel.
by Berarducci. Italian. This is the shrinked italian analog of the second part of Vol. 1 by Tourlakis.

InstitutionIndependent Model Theory.
by Diaconescu.

Proving classical theorems of social choice theory in modal logic.
by Ciná and Endriss.
Proof Theory

Proof Theory Foundations. Lecture 1,2,3,4.
by Frank Pfenning at Oregon Programming Languages Summer School 2012, University of Oregon.

Logical Frameworks  A brief introduction.
by Frank Pfenning.
Foundational aspects of Type Theory

Intuitionistic type theory.
by MartinLöf.

Semantics for type theory.
by Streicher.
Philosophy of Language

Word and object.
by Quine.

Naming and Necessity.
by Kripke.

Bibliography on Philosophy of Language.
by The London Philosophy study guide.

Bibliography on Logic and Metaphysics.
by The London Philosophy study guide.
Mathematical Stories

Indiscrete Thoughts.
by Rota.

A Mathematical Autobiography.
by Mac Lane.

A Mathematician's Apology.
by Hardy.

Adventures of a Mathematician.
by Ulam.

The Apprenticeship of a Mathematician.
by Weil.

by Bass, Cartan, Freyd, Heller, and Mac Lane in Notices of the AMS (1998).

An Interview with F. William Lawvere.
Bulletin of the International Center for Mathematics of Coimbra.

Editorial by Johnstone in Journal of Pure and Applied Algebra 116 (1997).

by Adámek and Paseka in Mathematica Bohemica, Vol. 132 (2007).

by Anita Burdman Feferman.
Mathematical Readings

The "What is...?" column of the AMS.
by the AMS news, gently indexed by Armin Straub.

Ten lessons I wish I had been taught.
by Rota.

Books about history of recent mathematics.
by the MathOverflow community.

How can a mathematician handle the pressure to discover something new?
by the MathOverflow community.

When is one 'ready' to make original contributions to mathematics?
by the MathOverflow community.

by Stan Brown.