n the first part of this thesis, we deal with homomorphisms from C*-algebras. Let A be (1) an AW*-algebra, (2) a unital C*-algebra with Hausdorff primitive spectrum, or (3) a C*-algebra which arises through an extension of a commutative C*-algebra by an elementary one. Then every homomorphism from A onto a dense subalgebra of a semisimple Banach algebra and every epimorphism from A onto a Banach algebra is continuous. Moreover, the singular part of every discontinuous homomorphism from A is accessible to further structural analysis. We prove a structure theorem for homomorphisms from A, which fully generalizes the classical Bade-Curtis theorem for homomorphisms from commutative C*-algebras. We give an example of a discontinuous homomorphism from a liminal, separable C*-algebra for which this structure theorem fails to hold.;In the second part, we investigate homomorphisms from group algebras. Let G be an (FIA) ;In the third and last part of the thesis, we are concerned with range inclusion problems for (possibly unbounded) derivations on Banach algebras. First, we give a characterization of all derivations on a Banach algebra that map into the radical; this extends earlier joint work by M. Mathieu and the author. Further, we deal with the question if for a derivation D on a Banach algebra A and an element
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