An important but under-studied problem in location analysis is the so called connected facility location problem. In such a problem, each facility is connected with other facilities by a certain network structure and the problem seeks to optimize facility locations so that the total costs including facility connection cost are minimized. Although its applications are seen in a number of network design domains including retail, telecommunication and public transportation, this problem is quite challenging to solve mathematically. In this dissertation, we study the connected facility location problem when both the demand set and the feasible set are continuous. We first introduce the continuous connected facility location problem and perform an asymptotic analysis to the problem. We then introduce a constant factor approximation algorithm for the problem and provide worst case analysis for the algorithm. We extend our analysis to a generalized connected facility location problem where the backbone network takes several different configurations and give an asymptotic analysis and an algorithmic analysis for each configuration. We finally discuss generalizations of our model for alternative cost models and multilevel networks.