An M x n matrix C is called Mange matrix if c" + cTT < clr + cr, for all 1 < i -c r < rn, 1 < j < s d n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.