In this thesis, we study obstructions that arise in problems originating from geometry and topology. For us, these obstructions are cohomological and, aside from computing them to obtain solutions to our problems, we aim to understand the extent to which they govern the behavior of the underlying problems. Our results include bisections of mass assignments by parallel hyperplanes on Euclidean vector bundles, complex analogues of the (colored) Tverberg--Vrećica conjecture, a solution to the Goodman-Pollack problem of determining the necessary and sufficient condition for the existence of k-transversals to a family of convex sets in a Euclidean space, and a necessary and sufficient cohomological condition for splitting off line bundles from a vector bundle.
