The problem of estimating a set S from a random sample of points arises in connection with some applications in statistical quality control, clustering, image analysis and statistical learning. This problem can be established in a more formal way as the problem of estimating the support of an absolutely continuous probability measure P from n independent observations drawn from P. So, the goal here is to estimate a set, not a parameter or a function. Assuming that the set of interest belongs to a certain family of sets can be useful in order to find an efficient estimation method. The case where S is assumed to be convex has received a special attention. If we assume that S is the support of the distribution which generates the sample points, there is a quite obvious estimator: the convex hull of the sample, that is, the smallest convex set which contains the sample. However, if S is not convex, the convex hull of the sample can be a bad choice. In this talk support estimation under the assumption that the set satisfies a much more flexible shape restriction, which is named alpha-convexity, will be presented. It will be showed that the new estimator can achieve, in a much more general setting, the same convergence rate as the convex hull. Support estimation is also connected to another interesting problem: the estimation of certain geometric characteristics of the set such as the volume or the surface area. It seems natural to think that the volume or the surface area of a good support estimator should provide good approximations of these geometrical quantities. Here we analyze the problem of boundary length estimation when it is assumed that the set is alpha-convex. Joint work with Beatriz Pateiro (Universidad de Santiago de Compostela), Antonio Cuevas (Universidad Autónoma de Madrid) and Ricardo Fraiman (Universidad de San Andres).