  We study the boundary regularity of almost minimal and quasiminimal sets that
satisfy sliding boundary conditions. The competitors of a set $E$ are defined
as $F = \varphi_1(E)$, where $\{ \varphi_t \}$ is a one parameter family of
continuous mappings defined on $E$, and that preserve a given collection of
boundary pieces. We generalize known interior regularity results, and in
particular we show that the quasiminimal sets are locally Ahlfors-regular,
rectifiable, and some times uniformly rectifiable, that our classes are stable
under limits, and that for almost minimal sets the density of Hausdorff measure
in balls centered on the boundary is almost nondecreasing.
