The primary goal of this talk is to introduce two equivalent definitions of algorithmically random sequences, one given in terms of a specific collection of effective statistical tests (known as Martin-Löf tests) and another given in terms of initial segment complexity (i.e., Kolmogorov complexity). I will explain how these definitions can be generalized to hold for various computable probability measures on Cantor space, and if time permits, I will discuss recent work with Rupert Hölzl and Wolfgang Merkle in which we study the interplay between (i) the growth rate of the initial segment complexity of sequences random with respect to some computable probability measure and (ii) certain properties of this underlying measure (such as continuity vs. discontinuity).  No background in algorithmic randomness will be assumed.