Let G be a linear algebraic group. The Dynkin variety ßx of an element x of G is the fixed point set of x on the variety ß of all Borel subgroups of G. We show that all irreducible components of this variety have the same dimension, and that ßx is connected if x is unipotent. Suppose now that G is reductive (but not necessarily connected) and that x is unipotent. We generalize an inequality linking dim ßxand dim Zꓖ (x) and some results on the action of A0(x) on the set S(x) of all irreducible components of ßx where A0(x) is the group of components of ZGo(x). We consider also regular and sub-regular elements in non-connected reductive groups. For classical groups we get a combinatorial description for S(x) and the action of A0(x) on S(x) and a formula for dim ßx We generalize to non-connected reductive groups a theorem of Richardson which associates to each conjugacy class of parabolic subgroups of G a unipotent class of G and for classical groups we get a combinatorial description of this map. There is also some material on unipotent classes in arbitrary reductive groups.