<p>This paper deals with developing a general spectral theory for only metrizable fuzzy normed algebras, whose topology is determined by functionals that may lack subadditivity. There are introduced the notions of fuzzy spectral radius, fuzzy boundedness radius, and fuzzy regular elements, and classical spectral results from Banach and locally convex algebras to this setting are extended. There are described fuzzy normed algebras induced by two strict t-norms and provide explicit examples, for which it is computed the fuzzy spectral radius and it is established the domain of fuzzy convergence for the Neumann series. A characterization of the fuzzy Waelbroeck resolvent set of regular elements is also given. As an application, the fuzzy Fourier transform on these algebras is investigated, proving to be a generalization of the classical transform to contexts governed by fuzzy rather than classical constraints.</p>