Let \(F\) be a field and \(G\) be a subgroup of the multiplicative group of \(F\) so that \(g\in G\) implies \(-g\in G\). A matroid representable over \(F\) by a matrix \(A\) is a \((G,F)\) matroid if all the non-zero subdeterminants of \(A\) are in \(G\). This generalization of regular (or totally unimodular) matroids led the authors to the study of matroids over partial fields (where the sum of two elements may not be defined). The class of these matroids is shown to be closed under standard matroid operations including dual, taking minors, direct sum and 2-sum. Relations to Dowling geometries are also studied.