The problem of hidden variables is examined in the axiomatic formulation of quantum mechanics based on the algebra of observables. After a brief introductory survey of the earlier investigations, we first investigate the structure ofC ^*-algebras which allow dispersion-free positive linear functionals. The result obtained is a direct generalization of the well-known result of von Neumann concerning the hidden variables. In the next Section, we assume, as before, that the observables form the Hermitian elements of aC ^*-algebra. But we now relax the requirement on «states» and allow the so-called monotone-positive functionals (which are not necessarily linear) to represent states. It is then shown that even when such generalized states are allowed, a system admits hidden variables only if its algebra of observables is Abelian;i.e., only if all observables are mutually compatible. In another Section, we investigate the question of hidden variables under the assumption that the observables, instead of forming aC ^*-algebra, have a certain more general algebraic structure.