  We discuss and develop some connections between analysis on singular spaces
and operator algebras, as presented in my sequence of four lectures at the
conference "Noncommutative geometry and applications," Frascati, Italy, June
16-21, 2014. Therefore this paper is mostly a survey paper, but the
presentation is new, and there are included some new results as well. In
particular, Sections 3 and 4 provide a complete short introduction to analysis
on noncompact manifolds that is geared towards a class of manifolds--called
"Lie manifolds"--that often appears in practice. Our interest in Lie manifolds
is due to the fact that they provide the link between analysis on singular
spaces and operator algebras. The groupoids integrating Lie manifolds play an
important background role in establishing this link because they provide
operator algebras whose structure is often well understood. The initial
motivation for the work surveyed here--work that spans over close to two
decades--was to develop the index theory of stratified singular spaces.
Meanwhile, several other applications have emerged as well, including
applications to Partial Differential Equations and Numerical Methods. These
will be mentioned only briefly, however, due to the lack of space. Instead, we
shall concentrate on the applications to Index theory.
