In this talk we discuss questions related to invertibility of random matrices, and the estimates for the smallest singular value. We outline the main results: an optimal small-ball behavior for the smallest singular value of square matrices under mild assumptions, and an essentially optimal small ball behavior for heavy-tailed rectangular random matrices. We make several remarks and outline some examples. We then explain the relation between such estimates and net constructions, and outline our main result in regards to existence of a net around the sphere with good properties. If time permits, we outline two more implications of this result.