This chapter is dedicated to counting partitions of sets and partitions of sets into cycles, and also introduces Stirling numbers and Bell numbers. As an application of the concepts discussed here we state Faa di Bruno chain rule for the n-th derivative of a composite of n-times differentiable functions on \(\mathbb R\). In the last section we discuss Eulerian numbers and as an application we solve the famous problem of the Smith College diplomas, and we establish some notable identities like Worpitzky’s formula.