Data on manifolds are emerging more and more frequently in various areas of modern research. Naturally, these demand proper probability distributions for modeling, prediction and statistical inference. In this talk, starting from ad hoc methods, a variety of general methods will be exploited to present novel statistical distributions on manifolds, e.g. circle, torus, cylinder, sphere, disc and their multivariate generalizations. The techniques are drawn from Fourier analysis, Complex analysis, functional analysis, stereographic projections, characteristic functions, Lie algebra, etc. Applications to modern real-life data are also given.