Let $\A$ be a $\Complex$-algebra, $\delta$ be a derivation on $\A$ and $\M$ be a left $\A$-module. A linear map $\tau : \M \rightarrow\M$ is called a generalized derivation relative to $\delta$ if $\tau(am)=a\tau(m)+\delta(a)m$ $(a\in\A,\ m\in\M)$. In this article first we study existence of generalized derivations. In particular we show that free modules and projective modules always have nontrivial generalized derivations relative to nonzero derivations of $\A$. Then we investigate invariance of prime submodules under generalized derivations. Specifically we show that every minimal prime submodule of $\M$ is invariant under every generalized derivation. Moreover we obtain analogs of Posner's theorem for generalized derivations. In the case that $\A$ is a Banach algebra and $\M$ is a Banach left $\A$-module, we study existence of continuous generalized derivations and automatic continuity of generalized derivations.
