  An approach to analysis on path spaces of Riemannian manifolds is described.
The spaces are furnished with `Brownian motion' measure which lies on
continuous paths, though differentiation is restricted to directions given by
tangent paths of finite energy. An introduction describes the background for
paths on ${\mathbb R}^m$ and Malliavin calculus. For manifold valued paths the
approach is to use `It\^o' maps of suitable stochastic differential equations
as charts . `Suitability' involves the connection determined by the stochastic
differential equation. Some fundamental open problems concerning the calculus
and the resulting `Laplacian' are described. A theory for more general
diffusion measures is also briefly indicated. The same method is applied as an
approach to getting over the fundamental difficulty of defining exterior
differentiation as a closed operator, with success for one \& two forms leading
to a Hodge -Kodaira operator and decomposition for such forms. Finally there is
a brief description of some related results for loop spaces.
