A weakly neighborly polyhedral map (w.n.p. map) is a 2-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We lay the foundation for an investigation of the w.n.p. maps of arbitrary genus. In particular we find all the w.n.p. maps of genus 0, we prove that for every g> the number of w.n.p. maps of genus g (orientable or not) is finite, and we find an upper bound for the number of vertices in a w.n.p. map of genus g>0. This upper bound grows as (4 g(2/3) when g→∞.