First-order viscous relativistic hydrodynamics has long been thought to be unsta- ble and acausal. This is not true; it is only with certain definitions of the hydrody- namic variables that the equations of motion display these properties. It is possible to define the hydrodynamic variables such that a fluid is both stable and causal at first order. This thesis does so for both uncharged and charged fluids, mostly for fluids at rest. Work has also been done in limited cases on fluids in motion. A class of stable and causal theories is identified via constraints on transport coefficients derived from linearized perturbations of the equilibrium state. Causality conditions are also derived for the full non-linear hydrodynamic equations. / Graduate