The most well known case of Teichmueller space is the one of complex tori, which yield a connected component
of $\mathcal{T}(T)$ isomorphic to an open set of a Grassmannian.
I shall present some recent results, the first joint with Pietro Corvaja, on the Teichmueller space
of Generalized Hyperelliptic manifolds. These are finite free quotients $X = T/G$ of a complex torus.
Using an algebraic result which says that a cristallographic group has a unique natural affine representation,
we can describe natural components for $\mathcal{T}(X)$, again open sets in suitable Grassmannian.
I shall then describe some joitn work with Gromadski , on surfaces of general type with
automorphisms acting trivially on cohomology, respectively trivially on rational but not on integral
cohomology. For irregular surfaces conjecturally there are no nontrivial automorphisms isotopic to
the identity.