This memoir is devoted to the study of the numerical treatment of source terms in hyperbolic conservation laws and systems. In particular, we study two types of situations that are particularly delicate from the point of view of their numerical approximation: The case of balance laws, with the shallow water system as the main example, and the case of hyperbolic equations with stiff source terms. In this work, we concentrate on the theoretical foundations of highresolution total variation diminishing (TVD) schemes for homogeneous scalar conservation laws, firmly established. We analyze the properties of a second order, flux-limited version of the Lax-Wendroff scheme which avoids oscillations around discontinuities, while preserving steady states. When applied to homogeneous conservation laws, TVD schemes prevent an increase in the total variation of the numerical solution, hence guaranteeing the absence of numerically generated oscillations. They are successfully implemented in the form of flux-limiters or slope limiters for scalar conservation laws and systems. Our technique is based on a flux limiting procedure applied only to those terms related to the physical flow derivative/Jacobian. We also extend the technique developed by Chiavassa and Donat to hyperbolic conservation laws with source terms and apply the multilevel technique to the shallow water system. With respect to the numerical treatment of stiff source terms, we take the simple model problem considered by LeVeque and Yee. We study the properties of the numerical solution obtained with different numerical techniques. We are able to identify the delay factor, which is responsible for the anomalous speed of propagation of the numerical solution on coarse grids. The delay is due to the introduction of non equilibrium values through numerical dissipation, and can only be controlled by adequately reducing the spatial resolution of the simulation. Explicit schemes suffer from the same numerical pathology, even after reducing the time step so that the stability requirements imposed by the fastest scales are satisfied. We study the behavior of Implicit-Explicit (IMEX) numerical techniques, as a tool to obtain high resolution simulations that incorporate the stiff source term in an implicit, systematic, manner.