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In this thesis we analyze consumption-investment problems with state-dependent discounting. We assume that the investor discounts his future utility by a non-exponential discount function which depends on his personal wealth and the state of the environment. This assumption makes the problem a so-called non-standard problem or time inconsistent problem, which has the consequence that the Bellman Optimality Principle does not hold. In order to find good time consistent strategies, we apply the concept of equilibrium strategies. We first analyze this problem in a discrete-time model, where we apply \textbf{Markov Decision Process (MDP)} methods to derive a characterization and calculation method for equilibrium strategies over the so-called extended Bellman equation. In continuous-time we analyze the problem on a diffusion market with deterministic coefficients. We derive a PIDE, the so-called extended Hamilton-Jacobi-Bellman equation, by taking the limit from discrete to continuous-time via the extended Bellman equation. In a verification theorem we prove that the equilibrium strategies in continuous-time can be characterized and computed over the extended Hamilton-Jacobi-Bellman equation. We compute closed-form solutions for equilibrium strategies in discrete and continuous-time for the three popular utility functions, logarithmic-, power- and exponential-utility in the case that the discount function is independent of the personal wealth of the investor.