We investigate notions of randomness in the space of continuous functions in Cantor space. A probability measure is given and a version of the Martin-Löf test for randomness is deŽned. Random continuous functions exist in the lower levels of arithmetic complexity, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. Any real y can be in the image of a random continuous function. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set.