Theoretical development and applications of graph signal processing (GSP) have attracted much attention. In classical GSP, the underlying structures are restricted in terms of dimensionality. A graph is a combinatorial object that models binary relations, and it does not directly model complex n-ary relations. A possible high-dimensional generalisaiton of graphs are simplicial complexes. They are a step between the constrained case of graphs and the general case of hypergraphs. In this paper, we develop a signal processing framework on simplicial complexes, such that we recover the traditional GSP theory when restricted to signals on graphs. We show some possible applications of our framework with high dimensional sensor networks.