Real forms and automorphism groups: some open questions

A real form of a complex variety X is a real variety whose complexification is isomorphic to X. Many varieties, for example curves or abelian varieties, are known to only have finitely many isomorphism classes of real forms. Recently, there has been some development in the field: now, there are examples of projective varieties (in fact, even surfaces) with infinitely many nonisomorphic real forms, and of affine surfaces with uncountably many. There are still some interesting questions once can pose, especially in connection with the automorphism group of the variety. I will give an overview of the field and relevant notions, and survey some open problems.