  $n$-to-$1$ mappings have wide applications in many areas, especially in
cryptography, finite geometry, coding theory and combinatorial design. In this
paper, many classes of $n$-to-$1$ mappings over finite fields are studied.
First, we provide a characterization of general $n$-to-$1$ mappings over
$\mathbb{F}_{p^m}$ by means of the Walsh transform. Then, we completely
determine $3$-to-$1$ polynomials with degree no more than $4$ over
$\mathbb{F}_{p^{m}}$. Furthermore, we obtain an AGW-like criterion for
characterizing an equivalent relationship between the $n$-to-$1$ property of a
mapping over finite set $A$ and that of another mapping over a subset of $A$.
Finally, we apply the AGW-like criterion into several forms of polynomials and
obtain some explicit $n$-to-$1$ mappings. Especially, three explicit
constructions of the form $x^rh\left( x^s \right) $ from the cyclotomic
perspective, and several classes of $n$-to-$1$ mappings of the form $ g\left(
x^{q^k} -x +\delta \right) +cx$ are provided.
