In the first part of my talk I will present the estimate of the Hausdorff dimension of the escaping set for elliptic functions (that is, the set of points that tend to infinity under iteration). It generalizes earlier results by Bergweiler, Kotus and Urbanski. The second part will be concerned with the estimates of the Hausdorff dimension of some subsets of the parameter space for families of elliptic functions. This is a joint work with J. Kotus.