  We study triple covers of K3 surfaces, following Miranda's theory of triple
covers. We relate the geometry of the covering surfaces with the properties of
both the branch locus and the Tschirnhausen vector bundle. In particular, we
classify Galois triple covers computing numerical invariants of the covering
surface and of its minimal model. We provide examples of non Galois triple
covers, both in the case in which the Tschirnhausen bundle splits into the sum
of two line bundles and in the case in which it is an indecomposable rank 2
vector bundle. We provide a criterion to construct rank 2 vector bundles on a
K3 surface $S$ which determine a non-Galois triple cover of $S$.
  The examples presented are in any admissible Kodaira dimension and in
particular we provide the constructions of irregular covers of K3 surfaces and
of surfaces with geometrical genus equal to 2 whose transcendental Hodge
structure splits in the sum of two Hodge structures of K3 type.
