  Path integrals are a ubiquitous tool in theoretical physics. However, their
use is sometimes hindered by the lack of control on various manipulations --
such as performing a change of the integration path -- one would like to carry
out in the light-hearted fashion that physicists enjoy. Similar issues arise in
the field of stochastic calculus, which we review to prepare the ground for a
proper construction of path integrals. At the level of path integration, and in
arbitrary space dimension, we not only report on existing Riemannian
geometry-based approaches that render path integrals amenable to the standard
rules of calculus, but also bring forth new routes, based on a fully
time-discretized approach, that achieve the same goal. We illustrate these
various definitions of path integration on simple examples such as the
diffusion of a particle on a sphere.
