We investigate notions of randomness in the space ${\mathcal {C}}[2^{\mathbb {N}}]$ of nonempty closed subsets of {0,1}ℕ. A probability measure is given and a version of the Martin-Lof Test for randomness is defined. Π02 random closed sets exist but there are no random Π01 closed sets. It is shown that a random closed set is perfect, has measure 0, and has no computable elements. A closed subset of $2^{{\mathbb N}}$ may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. This leads to some results on a Chaitin-style notion of randomness for closed sets.