Until a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time O ( n 2 3755 ) . Recently, a surge of activity by Stothers, Vassilevska-Williams, and Le Gall has led to an improved algorithm running in time O ( n 2 3729 ) . These algorithms are obtained by analyzing higher and higher tensor powers of a certain identity of Coppersmith and Winograd. We show that this exact approach cannot result in an algorithm with running time O ( n 2 3725 ) , and identify a wide class of variants of this approach which cannot result in an algorithm with running time O ( n 2 3078 ) ; in particular, this approach cannot prove the conjecture that for every 0"> 0 , two n n matrices can be multiplied in time O ( n 2+ ) . We describe a new framework extending the original laser method, which is the method underlying the previously mentioned algorithms. Our framework accommodates the algorithms by Coppersmith and Winograd, Stothers, Vassilevska-Williams and Le Gall. We obtain our main result by analyzing this framework. The framework is also the first to explain why taking tensor powers of the Coppersmith-Winograd identity results in faster algorithms.