In this work, we observed sequences of sets in metric and topological spaces, and we gave several defnitions of convergence of such sequences. First, we stated all the basic terms involving metric and topological spaces which were necessary for further observation. Initially, we defined the limit of a sequence (An)n∈N of subsets of X as the set A of all points that are limits of a sequence (xn)n∈N whose n-th term is an element of An. We called that limit the limit of the first kind. We generalized this definiton onto topological spaces, and then we generalized the sequences in that definition onto nets. One of the properties of the limit of the first kind helped us to make another definition, by which each point x ∈ X must satisfy d(x, An) → d(x, A). We called that limit the limit of the second kind. In the next step, the observed the Hausdorff metric dH, and we defined the limit of a sequence (An)n∈N of sets considering said distance as the non-empty closed set A which satisfies dH(An, A) → 0. We also observed convergence considering the Vietoris topology, and we gave a definition analogous to the definition of a limit of a sequence in a topological space. Lastly, we observed functions on sequences of sets. We proved that, if a function is continuous on each set in a sequence, it is not necessarily continuous on its limit of the first kind, or even in the interior of that set. Still, we gave one more definition of convergence of a sequence of sets, using pointwise convergence of characteristic functions of the given sets.