Polynomial methods and incidence theory The past decade has seen numerous major mathematical breakthroughs for topics such as the finite field Kakeya conjecture the cap set conjecture Erdöss distinct distances problem the joints problem as well as others thanks to the introduction of new polynomial methods. There has also been significant progress on a variety of problems from additive combinatorics discrete geometry and more. This book gives a detailed yet accessible introduction to these new polynomial methods and their applications with a focus on incidence theory. Based on the authors own teaching experience the text requires a minimal background allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front. The techniques are presented gradually and in detail with many examples warm up proofs and exercises included. An appendix provides a quick reminder of basic results and ideas Incidences and classical discrete geometry Basic real algebraic geometry in R² Polynomial partitioning Basic real algebraic geometry in Rd The joints problem and degree reduction Polynomial methods in finite fields The Elekes Sharir Guth Katz framework Constant degree polynomial partitioning and incidences in C² Lines in R³ Distinct distances variants Incidences in Rd Incidence applications in Rd Incidences in spaces over finite field Algebraic families dimension counting and ruled surfaces.