  We develop techniques to calculate an index for four dimensional
superconformal field theories. This superconformal index is counting BPS
operators which preserve only one supercharge. To calculate the superconformal
index we quantize the field theory on S^3 X R and show that the twisted theory
has an appropriate mass gap. This allows for the interactions to be switched
off continuously without the superconformal index being changed. We test those
techniques for theories which go through a non-trivial RG flow and for Seiberg
dual theories. This leads to the conjecture of some group/number theoretical
identities.
