Abstract: This paper is concerned with the nature of speedups. Let f be any recursive function. We show that there is no effective procedure for going from an algorithm for f to a significantly faster algorithm for f. On the other hand, there is an effective procedure for going from any algorithm to a faster algorithm, provided one has a bound on the size of the algorithm that does the computation faster for all inputs. If no bound on the size of the faster algorithm is known in advance, one can still obtain a pseudo speedup: This is a very fast algorithm which computes a variant of the function, one which differs from the original function on a finite number of inputs.