In three talks, I will describe aspects of the Local Lemma that have recently been uncovered by Moser & Tardos, Pegden, and David Harris and myself. As a running example, we will consider the following type of "graph transversal" problem introduced by Bollobas, Erdos and Szemeredi in the 1970s: given a graph $G = (V,E)$ and an integer $s$, for how small a $b$ can we guarantee that no matter how $V$ has been partitioned into blocks, each of size at least $b$, there is a way of choosing one vertex from each block such that the chosen vertices do not induce a clique on $s$ vertices?