In this paper we present a very general and unified theory of second-order optimality conditions for general optimization problems subject to abstract constraints in Banach spaces. Our results apply both to the scalar case and the multicriteria case. Our approach rests essentially on the use of a signed distance function for characterizing metric regularity of a certain multifunction associated with the problem. We prove variational results which show that, in a certain sense, our results are the best possible that one can obtain by using second-order analysis. We demonstrate how recently devised optimality conditions can be derived from our general framework, how they can be extended under weakened assumptions from the scalar case to the multiobjective case, and even how some new results also can be obtained.