We study best approximation problems with nonlinear constraints in Hilbert spaces. The strong "conical hull intersection property" (CHIP) and the "basic constraint qualification" (BCQ) condition are discussed. Best approximations with differentiable constraints and convex constraints are characterized. The analysis generalizes some linearly constrained results of recent works [F. Deutsch, W. Li, and J. Ward, J. Approx. Theory, 90 (1997), pp. 385--444; F. Deutsch, W. Li, and J. D. Ward, SIAM J. Optim., 10 (1999), pp. 252--268].