One of the central methods of arithmetic geometry is the study of algebraic varieties through the action of the Galois group on topological invariants of varieties such as cohomology and homotopy groups. An important question in this area is the problem of classification of representations of Galois groups that arise from cohomology of algebraic varieties. I will exhibit some of the known results in this direction and describe how working with families of algebraic varieties rather than a single variety leads to a substantially different behavior of the objects in question.