Many scientific and engineering applications require the solution of large systems of initial value problems arising from method of lines discretization of partial differential equations. For systems with widely varying time scales, or with complex physical dynamics, implicit time integration schemes are preferred due to their superior stability properties. However, for very large systems accurate solution of the implicit terms can be impractical. For this reason approximations are widely used in the implementation of such methods. The primary focus of this work is on the development of novel ``lightly-implicit'' time integration methodologies. These methods consider the time integration and the solution of the implicit terms as a single computational process. We propose several classes of lightly-implicit methods that can be constructed to allow for different, specific approximations. Rosenbrock-Krylov and exponential-Krylov methods are designed to permit low accuracy Krylov based approximations of the implicit terms, while maintaining full order of convergence. These methods are matrix free, have low memory requirements, and are particularly well suited to parallel architectures. Linear stability analysis of K-methods is leveraged to construct implementation improvements for both Rosenbrock-Krylov and exponential-Krylov methods. Linearly-implicit Runge-Kutta-W methods are designed to permit arbitrary, time dependent, and stage varying approximations of the linear stiff dynamics of the initial value problem. The methods presented here are constructed with approximate matrix factorization in mind, though the framework is flexible and can be extended to many other approximations. The flexibility of lightly-implicit methods, and their ability to leverage computationally favorable approximations makes them an ideal alternative to standard explicit and implicit schemes for large parallel applications. / Ph. D.