Let ⌫ be a finite index set and k Ն 1 a given integer. Let further ʕ [⌫]Յk be an arbitrary family of k element subsets of ⌫. Consider a (binomial) random subset ⌫ p of ⌫, where p ϭ ( p i : i ʦ ⌫) and a random variable X counting the elements of that are contained in this random subset. In this paper we survey techniques of obtaining upper bounds on the upper tail probabilities ‫(ސ‬X Ն ϩ t) for t Ͼ 0. Seven techniques, ranging from Azuma's inequality to the purely combinatorial deletion method, are described, illustrated, and compared against each other for a couple of typical applications. As one application, we obtain essentially optimal bounds for the upper tails for the numbers of subgraphs isomorphic to K 4 or C 4 in a random graph G(n, p), for certain ranges of p.