Algorithmic randomness was originally defined for Cantor space with the fair-coin measure. Recent work has examined algorithmic randomness in new contexts, in particular closed subsets of 2ɷ ( and ). In this paper we use the probability theory of closed set-valued random variables (RACS) to extend the definition of Martin-Löf randomness to spaces of closed subsets of locally compact, Hausdorff, second countable topological spaces. This allows for the study of Martin-Löf randomness in many new spaces, but also gives a new perspective on Martin-Löf randomness for 2ɷ and on the algorithmically random closed sets of  and . The first half of this paper is devoted to developing the machinery of Martin-Löf randomness for general spaces of closed sets. We then prove some general results and move on to show how the algorithmically random closed sets of  and  fit into this new framework.