  Two-to-one ($2$-to-$1$) mappings over finite fields play an important role in
symmetric cryptography. In particular they allow to design APN functions, bent
functions and semi-bent functions. In this paper we provide a systematic study
of two-to-one mappings that are defined over finite fields. We characterize
such mappings by means of the Walsh transforms. We also present several
constructions, including an AGW-like criterion, constructions with the form of
$x^rh(x^{(q-1)/d})$, those from permutation polynomials, from linear
translators and from APN functions. Then we present $2$-to-$1$ polynomial
mappings in classical classes of polynomials: linearized polynomials and
monomials, low degree polynomials, Dickson polynomials and
Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of
$2$-to-$1$ mappings over finite fields for constructions of bent Boolean and
vectorial bent functions, semi-bent functions, planar functions and permutation
polynomials. In all those respects, we shall review what is known and provide
several new results.
