We consider systems of partial differential equations of the form{uxt=F(u,ux,v,vx),vxt=G(u,ux,v,vx), describing pseudospherical (pss) or spherical surfaces (ss), meaning that, their generic solutions (u(x,t),v(x,t)) provide metrics, with coordinates (x,t), on open subsets of the plane, with constant curvature K=−1 or K=1. These systems can be described as the integrability conditions of g-valued linear problems, with g=sl(2,R) or g=su(2), when K=−1, K=1, respectively. We obtain characterization and also classification results. Applications of the theory provide new examples and new families of systems of differential equations, which also contain generalizations of a Pohlmeyer-Lund-Regge type system and of the Konno-Oono coupled dispersionless system. We provide explicitly the first few conservation laws, from an infinite sequence, for some of the systems describing pss.