The following thesis deals mainly with two different problems, both of which are related to the study of exceptional collections of sheaves. The first problem was motivated after a result in the study of D3-branes in String Theory established a connection between a Del Pezzo surface of degree 1 and the finite trihedral group Delta27 through a three block exceptional collection. We prove after understanding these collections in depth that this connection can only be extrapolated to two other cases: P2 is connected to Z3 and the smooth cubic surfaces in P3 are connected to Z3⊕ Z3 . The second problem was motivated as well by String Theory, and it consists of finding and describing the exceptional sheaves and collections on the singular reducible variety S=P2∪H 3 where P2∩H3 =P1 . The surface S became an object of study when a crepant resolution of the singularity of type ⅕(1, 1, 3) was found. More specifically, S is formed by the exceptional compact surfaces of the resolution. We provide insight in the study of the exceptionality of sheaves on S, finally showing that complete exceptional collections of invertible sheaves on S don't exist. We also show how to use mutations of exceptional pairs on S using the bounded derived category of sheaves Db(S).