Summary: New nonlinear FETI-DP (dual-primal finite element tearing and interconnecting) and BDDC (balancing domain decomposition by constraints) domain decomposition methods are introduced. In all these methods, in each iteration, local nonlinear problems are solved on the subdomains. The new approaches can significantly reduce communication and show a significantly improved performance, especially for problems with localized nonlinearities, compared to a standard Newton-Krylov-FETI-DP or BDDC approach. Moreover, the coarse space of the nonlinear FETI-DP methods can be used to accelerate the Newton convergence. It is also found that the new nonlinear FETI-DP and nonlinear BDDC methods are not as closely related as in the linear context. Numerical results for the \(p\)-Laplace operator are presented.