We introduce and solve stabilization problems for linear and nonlinear systems with state-dependent input delay. Since the state dependence of the delay makes the prediction horizon dependent on the future value of the state, which means that it is impossible to know a priori how far in the future the prediction is needed, the key design challenge is how to determine the predictor state. We resolve this challenge and establish closed-loop stability of the resulting infinite-dimensional nonlinear system for general non-negative-valued delay functions of the state. Due to an inherent limitation on the allowable delay rate in stabilization of systems with time-varying input delays, in the case of state-dependent delay, where the delay rate becomes dependent on the gradient of the delay function and on the state and control input, only regional stability results are achievable. For forward-complete systems, we establish an estimate of the region of attraction in the state space of the infinite-dimensional closed-loop nonlinear system and for linear systems we prove exponential stability. Global stability is established under a restrictive Lyapunov-like condition, which has to be a priori verified, that the delay rate be bounded by unity, irrespective of the values of the state and input. We also establish local asymptotic stability for locally stabilizable systems in the absence of the delay. Several illustrative examples are provided, including unicycle stabilization subject to input delay that grows with the distance from the reference position.