The level sets of probability density functions are of a considerable interest in many areas of statistics, and topological data analysis (TDA) in particular. In this talk we focus on the problem of recovering the homology of level sets from a finite sample. The main difficulty stems from the fact that even small perturbations to the estimated density function can generate a very large error in homology. In this talk we present an estimator that overcomes this difficulty and recovers the homology accurately (with a high probability). We discuss two possible applications of the proposed estimator. The first one is recovering the homology of a compact manifold from a noisy point cloud. The second application is recovering the persistent homology of the super level sets filtration. Finally, we show that similar methods can be used in the analysis of nonparametric regression models.