In this paper, we consider the problem of global stabilization via output feedback for a class of nonlinear systems which have been recently considered by many authors and are characterized by having nonlinear terms depending only on the output y. Our result incorporates many recent results. When static output feedback is considered, it is shown that the existence of an output control Lyapunov function, satisfying a suitable continuity property, is sufficient for constructing a continuous output feedback law u = k(y) which globally (or semiglobally) stabilizes the above class of systems, When dynamic output feedback is allowed, it is shown that the stabilization problem can be split into two independent stabilization subproblems : one is the corresponding problem via state feedback, and the other is the problem via output injection, From solving the two subproblems, one obtains two Lyapunov functions which, combined, give a candidate Lyapunov function for solving the output feedback stabilization problem, The proofs of our results give systematic procedures for constructing output feedback controllers, once two such Lyapunov functions are known, One can also consider the problem of output regulation and disturbance attenuation with global stability via measurement feedback and show that a similar "separation" condition holds.