Adaptation is characterized by the movement of a population toward a many-character optimum, movement that results in an increase in fitness. Here I calculate the rate at which fitness increases during adaptation and describe the curve giving fitness versus time as a population approaches an optimum in Fisher's model of adaptation. The results identify several factors affecting the speed of adaptation. One of the most important is organismal complexity--complex organisms adapt more slowly than simple ones when using mutations of the same phenotypic size. Thus, as Fisher foresaw, organisms pay a kind of cost of complexity. However, the magnitude of this cost is considerably larger than Fisher's analysis suggested. Indeed the rate of adaptation declines at least as fast as n-1, where n is the number of independent characters or dimensions comprising an organism. The present results also suggest that one can define an effective number of dimensions characterizing an adapting species.