  This memoir is a summary of recent work, including collaborations with Erik
van Erp, Christian Voigt and Marco Matassa, compiled for the "Habilitation \`a
diriger des recherches".
  We present various different approaches to constructing algebras of
pseudodifferential operators adapted to filtered and multifiltered manifolds
and some quantum analogues. A general goal is the study of index problems in
situations where standard elliptic theory is insufficient. We also present some
applications of these constructions.
  We begin by presenting a characterization of pseudodifferential operators on
filtered manifolds in terms of distributions on the tangent groupoid which are
essentially homogeneous with respect to the natural
$\mathbb{R}^\times_+$-action. Next, we describe a rudimentary multifiltered
pseudodifferential theory on the full flag manifold $\mathcal{X}$ of a complex
semisimple Lie group $G$ which allows us to simultaneously treat longitudinal
pseudodifferential operators along every one of the canonical fibrations of
$\mathcal{X}$ over smaller flag manifolds. The motivating application is the
construction of a $G$-equivariant $K$-homology class from the
Bernstein-Gelfand-Gelfand complex of a semisimple group.
  Finally, we discuss pseudodifferential operators on two classes of quantum
flag manifolds: quantum projective spaces and the full flag manifolds of
$SU_q(n)$. In particular, on the full flag variety of $SU_q(3)$ we obtain an
equivariant fundamental class from the Bernstein-Gelfand-Gelfand complex.
