Starting point of my talk are approximation algorithms for NP-hard problems in combinatorial optimization which are based on semidefinite programming (SDP), a recent and powerful method in convex optimization. One example is the theta number of Lovasz which provides an upper bound for the largest size of an independent set of finite graphs based on a solution of a semidefinite program. Many problems in extremal discrete geometry can be formulated in this way but for infinite geometric graphs. A famous example is the kissing number problem which goes back to a discussion of Newton and Gregory in 1692: What is the maximum number of non-overlapping unit balls that can simultaneously touch a central unit ball? To tackle this problem we generalize the theta number (and strengthenings based on SDP hierarchies) to infinite graphs which yields an infinite-dimensional semidefinite program. By using symmetries and tools from harmonic analysis it turns out that one can solve these semidefinite programs by computer giving the best known upper bounds in dimensions up to 24.