Let *G* be graph whose vertex set is partitioned into classes. When
does there exist a set *S* of vertices, consisting of one vertex from
each class, such that no two vertices of *S* are joined by an edge of
*G*? Such a set is called an *independent transversal* of *G*
with respect to the given vertex partition. It turns out that many
mathematical questions can be formulated by asking whether an
independent transversal exists in a particular graph with a particular
vertex partition.

We describe a number of different conditions that guarantee the
existence of an independent transversal in a given vertex-partitioned
graph. We also outline some applications of these results in various
different contexts.