Importance sampling is modified via {\it homotopy continuation} to improve the efficiency and success of the sampler. The homotopy will use a known distribution as a starting empirical importance sampling distribution and generate a continuous schedule which culminates with the target distribution. The focus is the estimation of the normalization constant of the target distribution. The homotopy method is extended to a Bayesian setting, for stationary and time dependent posterior distributions. The numerical implementation uses a combination of sample averages, with sampling parameter N, and homotopy stages M, where M is typically a small number. The algorithm replaces homotopy stages for sampling steps, potentially resulting in a better or more efficient importance sampler. Numerical experiments suggest this is the case. The results also suggest that the method may improve the efficiency of the sampler by concentrating the samples in regions of greater impact.