My talk discusses two families of BGG sequences that come from representation theory. Generalizing the construction for domains in $\mathbb R^n$, I will discuss a construction of these sequences in the setting of Riemannian manifolds and of manifolds endowed with a volume-preserving, torsion-free linear connection on their tangent bundle. An interesting feature of this construction is that produces complexes under the assumption that the metric in question is conformally flat or the connection is projectively flat. There also are interesting applications of BGG sequences that are not complexes, which I will briefly discuss in the talk.