The main topic of this dissertation is to introduce the notion of minimal log stable maps, which gives a new compactification of the space of stable maps relative to a Cartier divisor. A stable map relative to a Cartier divisor D is a holomorphic map from a nodal algebraic curve to a projective variety with prescribed tangency multiplicities along the marked points with respect to D. Rather than using the expanded degenerations, we adopt the tool of logarithmic geometry in the sense of Kato-Fontaine-Illusie. We first define the notion of log stable maps over schemes by equipping both source curves and the target of the usual stable maps with log structures, and show that the stack of log stable maps over schemes is algebraic. The log structures on the stable maps allow us to keep track of the tangency conditions even if the underlying map is degenerated. However, the stack of log stable maps over schemes is too large, and fails to be of finite type. Minimality is introduced to select a smaller open substack of the stack of log stable maps. The stack of minimal log stable maps is shown to be proper and Deligne-Mumford. Furthermore, the stack with its natural minimal log structure represents the category of log stable maps over fine and saturated log schemes. The representability allows us to generalize our construction to the case of generalized Deligne-Faltings pairs. In particular, this covers the case of stable maps relative to a simple normal crossings divisor.This is in part a joint work with my advisor Dan Abramovich.