Conserved quasilocal quantities and general covariant theories in two dimensions

General matterless theories in 1+1 dimensions include dilaton gravity, Yang-Mills theory, as well as non-Einsteinian gravity with dynamical torsion and higher power gravity, and even models of spherically symmetric d = 4 general relativity. Their recent identification as special cases of ''Poisson-sigma models'' with a simple general solution in an arbitrary gauge allows a comprehensive discussion of the relation between the known absolutely conserved quantities in all those cases and Noether charges, notions of quasilocal ''energy-momentum.'' In contrast with Noether-like quantities, quasilocal energy definitions require some sort of ''asymptotics'' to allow an interpretation as a (gauge-independent) observable. Dilaton gravitation, although a little different in detail, shares this property with the other cases. We also present a simple generalization of the absolute conservation law for the case of interactions with matter of any type.