Reacting flow phenomena occurring in real engines are complicated as a result of turbulent flow, interaction with solid boundaries, and extreme thermodynamic conditions. In order to understand and simulate combustion phenomena under such conditions, there is a necessity to develop accurate chemical kinetic and molecular transport models. Measurement of fundamental flame properties using low-dimensional experiments while avoiding complications associated with turbulence, heat loss, etc is central to the model development process. These measurements form targets against which models have to be validated and their uncertainties constrained. Knowledge of laminar flame ignition, propagation, extinction, and dynamic response to external transient effects is also crucial to the development of low-order, physics-based models that are necessary to design and optimize engines; current computational resources are not sufficient to simulate engine phenomena from first principles. In this talk, I will discuss how detailed numerical simulations were used to investigate and understand fundamental combustion phenomena using low-dimensional configurations. The topics that will be discussed include 1) measurement of laminar flame speeds at conditions of high pressure and temperature with focus on correct interpretation of experimental data obtained from spherically expanding flames, 2) localized initiation of ignition in the thermal boundary layer adjacent to a cold wall for reacting mixtures that exhibit the distinct negative temperature coefficient behavior, and 3) effects of unsteady thermodynamic pressure rise as present in internal combustion engines on flame propagation.