What are the best books to learn Mathematics and Physics?

I don't know!

But here are some that I enjoy or that I believe might help. I will include books and lecture notes in topics that I studied and that I feel fairly knowledgeable about.

Key

green = read and recommended by me

red = difficult to read

pink = elementary

grey = classic

blue = lecture notes

Real and Complex Analysis

Introductory

• Modern Real Analysis by William P. Ziemer
• Understanding Analysis by Stephen Abbott
• Principles of Mathematical Analysis by Walter Rudin
• Analysis Now by Gert K. Pedersen

Advanced

• Analysis, Sobolev Spaces and Partial Differential Equations by Gerald B. Folland
• Real and Complex Analysis by Walter Rudin

Functional Analysis

• Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis
• Functional Analysis by Walter Rudin
• Introductory Functional Analysis with Applications by Erwin Kreyszig
• Introduction to Topology and Modern Analysis by George F. Simmons
• History of Functional Analysis by Jean Dieudonné
• A course in Functional Analysis by John B. Conway
• Nonlinear Functional Analysis by Klaus Deimling
• Functional Analysis by Peter D. Lax
• Real and Functional Analysis by Serge Lang

Measure Theory

Introductory

• Measure Theory by by D.H.Fremlin
• Lebesgue Measure on Euclidean space by Frank Jone
• An introduction to measure theory by Terence Tao

Advanced

• Measure Theory Volume I by V.I. Bogachev
• Measure Theory Volume II by V.I. Bogachev
• Measure Theory by Paul R. Halmos
• Gaussian Measures by V.I. Bogachev

Stochastic Calculus

• Introduction to Stochastic Calculus with Applications by Fima C. Klebaner
• Stochastic Differential Equations by Bernt Oksendal
• A first Course in Stochastic Processes by Samuel Karlin and Howard M. Taylor
• Introduction to Probability Models by Sheldon M. Ross
• Brownian Motion - An Introduction to Stochastic Processes by René Schilling and Lothar Partzsch

Stochastic Calculus Applied to Finance

• Stochastic Calculus and Finance by Steven E. Shreve
• Stochastic Calculus and Financial Applications by Michael Steele
• Mathematical Finance by Ernst Eberlein and Jan Kallsen
• Interest Rate Models-Theory and Practice by Damiano Brigo and Fabio Mercurio
• The Volatility Surface-A Practitioner's Guide by Jim Gatheral

Finance

• Options, Futures and Other Derivatives by John C. Hull
• Commodities and Commodity Derivatives by Hélyette Geman
• Corporate Finance by Jonathan B. Berk and Peter DeMarzo

Quantum Mechanics

• Quantum Mechanics-A Paradigms Approach by David H. McIntyre
• Introduction to Quantum Mechanics by David J. Griffiths
• Lectures on Quantum Mechanics by Steven Weinberg
• Quantum Theory for Mathematicians by Brian C. Hall
• Principles of Quantum Mechanics by R. Shankar
• Modern Quantum Mechanics by J.J. Sakurai
• Topics in Quantum Mechanics by David Tong

Quantum Field Theory

• What is a Quantum Field Theory?-A First Introduction for Mathematicians by Michel Talagrand
• Quantum Field Theory-A Tourist Guide for Mathematicians by Gerald B. Folland
• Quantum Field Theory and the Critical Phenomena by Jean Zinn-Justin
• An Introduction to Quantum Field Theory by Michael E. Peskin and Daniel V. Schroeder
• Quantum Field Theory by Mark Srednicki
• Quantum Field Theory and the Standard Model by Matthew D. Schwartz
• The Quantum Theory of Fields by Steven Weinberg
• Quantum Field Theory-A Modern Perspective by V. Parameswaran Nair
• Quantum Field Theory in a Nutshell by A. Zee
• The Global Approach to Quantum Field Theory by Bryce DeWitt
• Stochastic Quantization by Pout H. Damgaard and Helmuth Huffel
• Quantum Field Theory by David Tong
• Quantum Field Theory by Sidney Coleman
• Quantum Field Theory I by Arthur Hebecker
• Quantum Field Theory by Liam Fitzpatrick
• Quantum Field Theory by Timo Weigand
• 6 Lectures on QFT, RG and SUSY by Timothy J. Hollowood
• Quantum Field Theory II by David Skinner
• The Standard Model by David Tong

Stochastic Partial Differential Equations

• Stochastic Equations in Infinite Dimensions by Giuseppe Da Prato and Jerzy Zabczyk
• A Course on Rough Paths-With an introduction to regularity structures by Peter K. Friz and Martin Hairer
• Stochastic PDEs, Regularity Structures, and Interacting Particle Systems by Ajay Chandra and Hendrik Weber
• Paracontrolled distributions and singular PDEs Massimiliano Gubinelli, Peter Imkeller and Nicolas Perkowski