In this paper we characterize the maximal modular ideals of an evolution algebra A in order to describe its Jacobson radical, Rad(A). We characterize semisimple evolution algebras (i.e. those such that Rad(A) = {0})as well as radical ones. We introduce two elemental notions of spectrum of an element a in an evolution algebra A, namely the spectrum σ A (a) and the m-spectrum σ A m (a) (they coincide for associative algebras, but in general σ A (a) ⊆ σ A m (a), and we show examples where the inclusion is strict). We prove that they are non-empty and describe σ A (a) and σ A m (a) in terms of the eigenvalues of a suitable matrix related with the structure constants matrix of A. We say A is m-semisimple (respectively spectrally semisimple) if zero is the unique ideal contained into the set of a in A such that σ A m (a) = {0} (respectively σ A (a) = {0}). In contrast to the associative case (where the notions of semisimplicity, spectrally semisimplicty and m-semisimplicity are equivalent) we show examples of m-semisimple evolution algebras A that, nevertheless, are radical algebras (i.e. Rad(A) = A). Also some theorems about automatic continuity of homomorphisms will be considered.