Suppose computing a function f requires C computational resources under some model of computation. Does this mean that computing k copies of f requires a larger amount of resources? This type of question is well studied in complexity and is usually referred to as the "direct sum" question. In this work, we prove the first known direct sum lower bounds for randomized communication complexity. We show that if two parties need to communicate C bits to compute some joint function f of their inputs, then they must need at least C sqrt{k} bits to compute k copies of f. In the more restricted case of computing f on average under the uniform distribution (in fact our results apply to any product distribution), we give close to optimal bounds, showing that computing k copies is requires almost k times as much communication. Previously such bounds were only known for communication protocols with a bounded number of rounds, and even then only when the inputs come from a product distribution (Harsha, Jain, McAllester and Radhakrishnan'07).