This paper characterizes modular group algebras and restricted enveloping algebras satisfying a nonmatrix identity or a semigroup identity. Let \(K\) be a field of characteristic \(p>0\). Then it is shown that \(K[G]\) satisfies a nonmatrix identity if and only if (1) \(G\) has a normal subgroup \(A\) with \(G/A\) and \(A'\) both finite, and (2) \(G'\) is a \(p\)-group of finite exponent. Similarly, \(u(L)\) satisfies a nonmatrix identity if and only if (1) \(L\) has a restricted ideal \(A\) with \(L/A\) and \(A'\) both finite dimensional, and (2) \(L'\) is \(p\)-nil of bounded index. If \(K\) is an infinite field, then analogous results characterize group algebras and restricted enveloping algebras satisfying a semigroup identity. The paper also discusses the possibility that the augmentation ideal of \(K[G]\) or \(u(L)\) satisfies one of these two types of identities. The question in the case of semigroup identities seems to be rather delicate.