spherical designs are finite sets of points on the sphere with the property that the average of low-degree polynomials over the sphere coincides with the average over the finite set. these objects are very beautiful, very symmetric and have been studied since the 1970s. we use a completely new approach that replaces delicate combinatorial arguments with the a simple application of the heat equation; this approach improves the known results and extends to other manifolds. we also discuss some related issues in combinatorics, irregularities of distribution and fourier analysis.