Abstract: Convolution of functions and measures on Euclidean spaces, and their connections with sums of independent random variables and Levy processes, are well known. These notions can be naturally extended to Lie groups. Many of the basic results are essentially parallel to those on Euclidean spaces, but there are also many interesting features and results for the convolution on Lie groups that are not present for its counter part on Euclidean spaces. We will present some of these results, including the connection with Levy processes in Lie groups, the problem of embedding an infinitely divisible distribution in a convolution semigroup, and the convergence of convolution power of a distribution on a compact group.