The main result of this work is a simple characterization of the hyperstability of functional equations in a very general framework: for functions with the codomain endowed with a generalized homogeneous premetric. We also give some particularizations on Abelian groups equipped with generalized homogeneous norms, obtaining, among others, improvements of similar known results for Cauchy-type, Jensen-type equations, but also for equations having compound functions as solutions.