Extending the regularity theory of R. Schoen and K. Uhlenbeck [J. Differ. Geom. 17, 307-335 (1982; Zbl 0521.58021)] the author proves regularity of \(p\)-energy minimizing maps \(u:M^n \to S^k\) of an \(n\)-dimensional Riemannian manifold \(M^n\) into the Euclidean \(k\)-sphere \(S^k\). Regularity is obtained under conditions imposed on \(n,k\) and \(p\), and, for \(p\searrow 2\), these conditions reduce to the Schoen-Uhlenbeck regularity result [Invent. Math. 78, 89-100 (1984; Zbl 0555.58011)].