This paper presents asymptotically optimal algorithms for rectangular matrix transpose, FFT, and sorting on computers with multiple levels of caching. Unlike previous optimal algorithms, these algorithms are cache oblivious: no variables dependent on hardware parameters, such as cache size and cache-line length, need to be tuned to achieve optimality. Nevertheless, these algorithms use an optimal amount of work and move data optimally among multiple levels of cache. For a cache with size Z and cache-line length L where \math the number of cache misses for an \math matrix transpose is \math. The number of cache misses for either an n-point FFT or the sorting of n numbers is \math. We also give an \math-work algorithm to multiply an \math matrix by an \math matrix that incurs \math cache faults.We introduce an `ideal-cache' model to analyze our algorithms. We prove that an optimal cache-oblivious algorithm designed for two levels of memory is also optimal for multiple levels and that the assumption of optimal replacement in the ideal-cache model can be simulated efficiently by LRU replacement. We also provide preliminary empirical results on the effectiveness of cache-oblivious algorithms in practice.