Wonderful varieties are algebraic varieties, equipped with an action of a linear semisimple group $G$ and satisfying axioms inspired by the well known compactifications of symmetric spaces of De Concini and Procesi. They are a sort of generalization of complete $G$-homogenous spaces (i.e. partial flag varieties, if $G$ is the special linear group $SL(n)$). In the talk we will discuss some of their basic properties, and many examples. In particular, we will focus on recent results about their full automorphism groups, generalizing the results of Demazure (1974) about automorphism groups of complete $G$-homogeneous spaces.