  Opened up by early contributions due to, among others, H. Bohr, Hardy-Riesz,
Bohnenblust-Hille, Neder and Landau the last 20 years show a substantial
revival of systematic research on ordinary Dirichlet series $\sum a_n n^{-s}$,
and more recently even on general Dirichlet series $\sum a_n e^{-\lambda_n s}$.
This involves the intertwining of classical work with modern functional
analysis, harmonic analysis, infinite dimensional holomorphy and probability
theory as well as analytic number theory. Motivated through this line of
research the main goal of this article is to start a systematic study of a
variety of fundamental aspects of vector-valued general Dirichlet series $\sum
a_n e^{-\lambda_{n} s}$, so Dirichlet series, where the coefficient are not
necessarily in $\mathbb{C}$ but in some arbitrary Banach space $X$.
