The parabolic geometries are curved deformations of (real) homogeneous spaces G/P, such that the complexification $P^{\mathbb{C}} \subset G^L$ is a parabolic subgroup in a semisimple Lie group. The general ideas go back to Cartan's 'generalized spaces'. Examples involve the conformal, projective, almost quaternionic, and CR geometries. The first talk will present an introduction, survey of recent existence results, and a more detailed explanation of new techniques leading to an effective calculus for invariant operators. The explicit construction of curved analogues of the famous Bernstein-Gel'fand-Gel'fand resolutions for all these geometries (without any use of its representational theoretical version) will be presented as the first major application of our new approach. The second talk will focus on the parabolic geometries with irreducible tangent bundles. In this case, all the standard geometrical structures known from conformal Riemannian geometries admit nice general counterparts and, in particular, there is a more straightforward explicit construction of the operators from the above mentioned resolutions based on finite dimensional representation theory. It is remarkable that the closed formulae for all these operators of a given order do not depend on the choice of the structure groups.
