The goal of this work is to generalize the analysis of adaptive algorithms for finite element methods (FEM) and boundary element methods (BEM) from elliptic problems, satisfying the setting of the Lax-Milgram theorem, to certain classes of elliptic indefinite and nonlinear problems. For each problem class, based on an a-posteriori error estimator, we introduce an adaptive algorithm and prove that these algorithms do not only lead to linear convergence, but also guarantee optimal algebraic convergence behavior of the underlying error estimator. The thesis is split into two parts, where each part analyzes one specific problem class in an abstract framework. This general approach allows to formulate so-called axioms of adaptivity for the error estimator as well as the underlying mesh-refinement strategy, under which optimal algebraic convergence can be guaranteed. First, we consider indefinite and compactly perturbed elliptic problems. This problem class covers general diffusion problems with convection and reaction and, in particular, the Helmholtz equation. For a standard conforming FEM and BEM discretization by piecewise polynomials, usual duality arguments show that the underlying triangulation has to be sufficiently fine to ensure the existence and uniqueness of the Galerkin solution. Extending the abstract approach of existing works, we prove that adaptive mesh-refinement is capable of overcoming this preasymptotic behavior and eventually leads to convergence with optimal algebraic rates. Unlike previous works, one does not have to deal with the a-priori assumption that the initial mesh is sufficiently fine. Due to stabilizing effects, the adaptive algorithm can, in particular, overcome possibly pessimistic restrictions on the meshes. As an application of the abstract framework, we prove optimal algebraic convergence rates for adaptive FEM. Further, we show inverse estimates for the most important boundary integral operators associated with the Helmholtz equation, which generalizes the existing results for the Laplace equation to arbitrary wavenumbers. This allows us to give a first prove of optimal convergence rates for adaptive BEM for the Helmholtz equation. One particular strength of the boundary element methods is, that it allows for a higher-order point-wise approximation of the solution. As an application of the prior analysis, we generalize existing results for the elliptic case and prove optimal convergence behavior with respect to an a-posteriori computable bound for the point error of the Helmholtz equation. In the second part, we focus on nonlinear PDEs with strongly monotone operators. Unlike prior works, the analysis includes the iterative and inexact solution of the arising discrete nonlinear systems by means of the Picard iteration. We also consider an iterative PCG-solver for the invoked linear system in the computation of each Picard step. Using nested iteration, we show an improved linear convergence result as well as optimal algebraic convergence behavior of the underlying error estimator. Improving existing results, we also prove optimal convergence rates with respect to the cumulative computational costs of the adaptive algorithm.