The problem of finding the connected components of an undirected graph is one of the most basic in graph algorithms. It can be solved sequentially in linear time using graph search or in almost-linear time using a disjoint-set data structure. The latter solves the incremental version of the problem, in which edges are added singly or in batches on-line. With the growth of the internet, computing connected components on huge graphs has become important, and both experimentalists and theoreticians have explored the use of concurrency in speeding up the computation. We shall survey recent work. Even simple concurrent algorithms are hard to analyze, as we discuss. This work is joint with Sixue Liu of Princeton. The opinions expressed in this video do not necessarily reflect the views of the Heidelberg Laureate Forum Foundation or any other person or associated institution involved in the making and distribution of the video.