The topic of this thesis is approximation and online algorithms for an optimization problem known as Facility Location. This problem, or one of its many variants, arises as a sub problem in many practical applications, and is thus of significant importance in the field of Operations Research. Furthermore, it is also one of the most studied optimization problems in theoretical computer science with hundreds of research papers published during the last decades. In this thesis, we focus on the theoretical aspects of Facility Location by designing and analyzing approximation and online algorithms. Our algorithms deal with three distinct scenarios in which Facility Location occurs: (i) networks that are exposed to perpetual changes, (ii) wireless sensor networks with strong locality constraints, and (iii) distributed settings where the focus lies, first and foremost, on the quality of the computed approximation. Chapter 2 covers Scenario (i). It presents an online algorithm designed for a highly dynamic network where additional nodes are perpetually added. The difficulty here is that these new nodes' requests have to be handled efficiently without any knowledge of the network's future development. Scenario (ii) is considered in Chapter 3. Two distributed algorithms for wireless sensor networks are presented here. Due to the nodes' limited communication range, locality is of high importance in this scenario. Additional aspects like inaccurate measurement data, power consumption, and dynamics are also taken into account. Finally, Scenario (iii) is considered in Chapter 4. Our objective here is to distributedly compute a solution with an approximation ratio that is as close as possible to the best achievable ratio. In order to accomplish this, we allow, compared to Scenario (ii), a higher running time, but still require that the algorithm terminates in sub-linear time.