A CAT(0) space is a geodesic metric space having globally non positive curvature. In this talk, I will discuss combinatorial and algorithmic aspects of CAT(0) spaces associated with combinatorial objects (graphs, posets, etc), from applied mathematics points of view. Topics include: (1) The space of phylogenetic trees (Billera, Holmes, Vogtmann 2001), and Owen-Provan algorithm (Owen, Provan 2011) computing the geodesic between two phylogenetic trees, via parametric network flow. (2) Orthoscheme complexes associated with posets (Brady-McCammond 2010), classes of lattices/semilattices having CAT(0) orthoscheme complexes, and their application to sub-modular optimization.