We review how to bound the error between the unknown weak solution of a PDE and its numerical approximation via a fully computable a posteriori estimate. We focus on approximations obtained at an arbitrary step of a linearization (Newton-Raphson, fixed point, ...) and algebraic solver (conjugate gradients, multigrid, domain decomposition, ...). Identifying the discretization, linearization, and algebraic error components, we design local stopping criteria which keep them in balance. This gives rise to a fully adaptive inexact Newton method. Numerical experiments are presented in confirmation of the theory.