The notion of a cluster set is due originally to Painleve who introduced it as a means of studying the singularities of solutions of certain types of ordinary differential equations. The main stimulus to the development of the theory has come from the study of meromorphic functions in the neighborhood of an isolated essential singularity. The theorems of Weierstrass and Picard form the first important results in this area. Before stating these theorems we define the cluster set C(f,zo) at a point zo for a function f(z) defined in a domain D, zo ∈ D¯, where D¯ denotes the closure of D.