Early reservoir models (e.g., black-oil reservoir models) were typically based upon rigorous mass balance equations for key species (oil, water, and gas) but only used approximate phase equilibria (e.g., no oil dissolved in the water phase) and/or neglected energy balances. By 1981, reservoir simulation had reached a level of maturity to warrant the first Society of Petroleum Engineers (SPE) Comparative Solutions Project on 3-D Black Oil Reservoir Simulation (Odeh 1981), in which seven different companies participated in a model-development exercise. To date, there have been ten separate comparative solution projects sponsored by the SPE with topics that include three-phase behavior, steam injection, horizontal wells, and effective grid generation and up-scaling techniques. These Comparative Solutions Project papers are useful for readers new to reservoir simulation or those simply interested in learning more about challenging issues in this area. By 2001, reservoir simulation had reached a point where advanced concepts such as dual-porosity models, rigorous phase behavior, energy-balance considerations, fully-implicit time stepping with Newton's method to solve the reservoir model equations at each time step, iterative linear solvers, finite difference, and/or analytical Jacobian matrices (to name a few) were available as modeling components. Today, there remains considerable oil in place (OIP) in many reservoirs that are either in current operation or have been shut down (often with infrastructure remaining in place). There are also large amounts of fossil fuels in heavy oil, oil sands, and deep-sea reservoirs, but these hydrocarbons are more challenging and more costly to produce. An increase in production of a standard oil field of just 1% can represent a $25B opportunity.