Adaptive finite element methods (AFEMs) have been successfully used in the numerical solution of partial differential equations (PDEs) to improve rates of convergence compared to standard FEM employing uniform meshes. However, due to the cumulative nature of adaptive mesh-refinement, one strives to achieve optimal complexity, i.e., optimality with respect to the overall computational cost (and hence the total elapsed time). The core question is the design of appropriate iterative solvers used within AFEM balancing the different error components, e.g., in the case of a nonlinear PDE, consisting of the discretization error, linearization error, and algebraic error. The numerical analysis of such algorithms relies on the cornerstone notion of R-linear convergence of an appropriate quasi-error quantity. It yields that, independently of the chosen adaptivity parameters, convergence is guaranteed and, moreover, the rates with respect to the number of degrees of freedom coincide with those with respect to the computational complexity. Numerical experiments highlight the theory and emphasize the practical relevance and gain of adaptivity with iterative solvers for numerical simulations with optimal complexity.