  We initiate the axiomatic study of affine oriented matroids (AOMs) on
arbitrary ground sets, obtaining fundamental notions such as minors,
reorientations and a natural embedding into the frame work of Complexes of
Oriented Matroids. The restriction to the finitary case (FAOMs) allows us to
study tope graphs and covector posets, as well as to view FAOMs as oriented
finitary semimatroids. We show shellability of FAOMs and single out the FAOMs
that are affinely homeomorphic to $\mathbb{R}^n$. Finally, we study group
actions on AOMs, whose quotients in the case of FAOMs are a stepping stone
towards a general theory of affine and toric pseudoarrangements. Our results
include applications of the multiplicity Tutte polynomial of group actions of
semimatroids, generalizing enumerative properties of toric arrangements to a
combinatorially defined class of arrangements of submanifolds. This answers
partially a question by Ehrenborg and Readdy.
