It is a key observation, made by Kapranov, Mikhalkin, Sturmfels, and others, that relevant geometric problems have a combinatorial core. This lead to the invention of tropical geometry as something like a "piece-wise linear shadow of algebraic geometry". Specializing tropical algebraic geometry to the linear case leads to the subject of max-plus linear algebra. This field, rooted in optimization, is much older than tropical geometry. It is linked with signal processing, functional analysis and other areas. The goal of this talk is to give examples of simple combinatorial concepts which evolve into theorems which are interesting for pure as well as applied mathematics.