A set of lines in $R^n$ is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in $R^n$, using semidefinite programming to improve the upper bounds on this quantity. Improvements are obtain in dimensions $23\leq n \leq 136$. In particular, we show that the maximum number of equiangular lines in $R^n$ is 276 for all $24 \leq n \leq 41$ and 344 for $n=43$. This provides a partial resolution of the conjecture set forth by Lemmens and Seidel (1973). We also study the existence problem for tight spherical designs of harmonic index T. We prove the nonexistence of tight {8,4} designs by using the theory of elliptic diophantine equations, and the semidefinite programming method of eliminating some 2-angular systems for small dimensions.