Abstract: This master thesis consists of two independent parts. In the first one, basics of Hermitian geometry are reviewed in order to prove a result by Paul Gauduchon stating the existence and uniqueness of a Hermitian connection of minimal torsion on a given almost-Hermitian manifold. In the second one, we study spinors lying in the kernel of a Penrose-type operator on conformal spin manifolds according to Volker Buchholz's master thesis.