  Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of
characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there
is an idempotent homomorphism (a {\it retraction}, or {\it projection})
$\varphi: K[x, y] \to K[x, y]$ such that $\varphi(K[x, y]) = R$. The presence
of other, equivalent, definitions of retracts provides several different
methods of studying them, and brings together ideas from combinatorial algebra,
homological algebra, and algebraic geometry. In this paper, we characterize all
the retracts of $ K[x, y]$ up to an automorphism, and give several applications
of this characterization, in particular, to the well-known Jacobian conjecture.
Notably, we prove that if a polynomial mapping $\varphi$ of $K[x,y]$ has
invertible Jacobian matrix {\it and } fixes a non-constant polynomial, then
$\varphi$ is an automorphism.
