The aim of these lecture notes is to describe algebraic varieties on which an algebraic group acts and the orbit structure is simple. We begin by presenting fundamental results for homogeneous varieties under (possibly nonlinear) algebraic groups. Then we turn to the class of log homogeneous varieties, for which the orbits are the strata defined by a divisor with normal crossings. In particular, we discuss the close relationship between log homogeneous varieties and spherical varieties, and we survey classical examples of spherical homogeneous spaces and their equivariant completions.