An oriented hypergraph is an oriented incidence structure that is a hypergraphic extension of signed graphs. I will introduce the main concepts and obstacles relevant to the classification of circuits of {0,+1,-1}-matrices, and discuss recent progress towards a complete circuit classification of representable matroids. As time permits, I will also discuss related work on the hypergraphic laplacian, as well as a new decomposition result for contraction in representable matroids.