In most of this book we have studied additive combinatorics problems in an ambient group Z, relying primarily on the additive structure of Z (as manifested for instance in the Fourier transform). However, in many cases the ambient group is in fact a field F, and thus supports a number of special functions, in particular polynomials. One can then use tools from algebraic geometry to exploit these polynomial structures; this is known as the polynomial method. One of the primary ideas here is to interpret an additive set (e.g. a sum set A + B) as the zero locus of one or more polynomials, possibly in several variables. One can then hope to control the size of such sets using results from algebraic geometry about the number and distribution of zeroes of polynomials. The most familiar example of such a theorem is the statement that a polynomial P(t) of one variable with degree d in a field F can have at most d zeroes; however for most applications we will need to study the zero locus of polynomial(s) in many variables. In this chapter we present four related tools and techniques from algebraic geometry which allow one to control such a zero locus. The first is the powerful combinatorial Nullstellensatz of Alon (Theorem 9.2), which asserts that the zero locus of a polynomial P(t1, …, tk) cannot contain a large box S1 × … × Sk if a certain monomial coefficient of P is non-vanishing; this is particularly useful for obtaining lower bounds on the size of restricted sum sets and similar objects.