Laser additive manufacturing refers to advanced laser-based fusion processes used to add features to an existing part, as a means for more efficient part manufacture or part repair. In this research, the control of steady-state and transient melt pool size in laser additive manufacturing processes is studied. Melt pool size is a key process characteristic that must be controlled to allow the precise deposition of complex features. For the case of steady-state conditions, numerical simulations are used to construct non-dimensional plots (termed process maps) that quantify the effects of changes in part height, laser power, deposition speed and part preheating on melt pool size. Results originally developed for a small-scale process are extended to large-scale processes, with conclusions on the role of process size on process robustness. An understanding of transient changes in melt pool size is an important part of efforts to control melt pool size in real time, via thermal imaging or other feedback control systems. The process map approach applied to the steady-state problem is also extended to the study of transient changes in melt pool size due to a step change in laser power or velocity. The range of thermal response times is identified for practical combinations of process variables. Process thermal response times set a lower bound for the response time of thermal feedback control systems. This research is being performed in tandem with research at Wright State University and Ohio State University on relationships between process variables and deposit microstructure.