The Langlands correspondence, coupled with other fundamental
conjectures such as that of Fontaine-Mazur, predicts a "basic
triangle" of correspondences between (1) certain automorphic
representations, (2) certain Galois representations and (3) pure
motives. We shall begin by briefly saying something about these three
classes of objects and how they are expected to correspond to one
another. We will introduce and motivate the regular/irregular
dichotomy for the objects (1)-(3), explaining why the irregular case
is at the same time more interesting and more difficult. The remainder
of the talk will focus on the arrow (1)-->(2) in the irregular case.
There are essentially two known techniques for attacking the latter
problem: Exploiting the geometry of Shimura varieties and Langlands'
Functoriality Principle. We will present recent results and work in
progress using each of the two techniques. Our geometric results are
joint work with J.-S. Koskivirta, following earlier joint work with