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For each simple Lie algebra g consider the corresponding affine vertex algebra V(g) at the critical level.The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago.The field corresponding to any element of the Feigin-Frenkel center is a Sugawara operator.We give explicit formulas for generators of the center and hence for Sugawara operators for all classical Lie algebras g.Our approach is based on the Schur-Weyl duality and leads to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g) and to explicit formulas for higher Gaudin Hamiltonians.