In this thesis, we are concerned with the construction and analysis of high-order exponential integrators for the time discretization of large systems of stiff differential equations. The construction of exponential integrators heavily relies on the knowledge of the order conditions, which are available in the literature up to order four. We introduce a novel approach for deriving the stiff order conditions for two popular classes of exponential integrators, namely exponential Runge-Kutta and exponential Rosenbrock methods. Based on this new approach, we obtain stiff order conditions for such exponential methods of order up to five. These conditions allow us to construct methods of order five. As examples, we present fifth-order exponential Runge-Kutta and exponential Rosenbrock methods. For particular problems the new order conditions can be simplified. It is then even possible to construct an exponential Rosenbrock method of order five with three stages only. The error analysis is performed in an abstract Banach space framework of strongly continuous (or analytic) semigroups. Convergence results are proven independently of the stiffness of the problem. Moreover, we investigate an alternative approach for deriving the stiff order conditions by extending the well-known concept of B-series to exponential integrators. More precisely, we derive a stiff B-series theory based on the variation-of-constants formula. With the help of this theory, the stiff order conditions for such exponential methods of arbitrary order can be obtained in a simple way from a set of recursively defined trees. The implementation of new integrators and the comparison with existing methods are carried out by using MATLAB. In particular, numerical examples for semilinear parabolic PDEs in one and two space dimensions are given to demonstrate the efficiency of the new integrators.