The present constitutes the lecture notes from a mini course at the Summer School ”Structures in Lie Representation Theory” from Bremen in August 2009.
The aim of these lectures is to describe algebraic varieties on which an algebraic group acts and the orbit structure is simple. The methods that will be used come from algebraic geometry, and representation theory of Lie algebras and algebraic groups.
We begin by presenting fundamental results on homogeneous varieties under (possibly non-linear) algebraic groups. Then we turn to the class of log homogeneous varieties, recently introduced in  and studied further in ; here 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.