In linear algebra, a **multilinear map** is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function

- $f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}$

where $V_{1},\ldots ,V_{n}$ and $W$ are vector spaces (or modules over a commutative ring), with the following property: for each $i$, if all of the variables but $v_{i}$ are held constant, then $f(v_{1},\ldots ,v_{n})$ is a linear function of $v_{i}$.

A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, a multilinear map of *k* variables is called a *k*-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.

If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating *k*-linear maps. The latter coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.