Facility Location problems ask to place facilities in a way that optimizes a given objective function so as to provide a service to all clients. These are one of the most well-studied optimization problems spanning many research areas such as operations research, computer science, and management science. Traditionally, these problems are solved with the assumption that clients need to be served by one facility each. In many real-world scenarios, it is very likely that clients need a robust service that requires more than one facility for each client. In this paper, we capture this robustness by exploring a generalization of Facility Location problems, called Multi-Facility Location problems, in the online setting. An additional parameter k, which represents the number of facilities required to serve a client, is given. We propose the first online algorithms for the metric and non-metric variants of Multi-Facility Location and measure their performance with competitive analysis, the standard to measure online algorithms, in the worst case, in which the cost of the online algorithm is compared to that of the optimal offline algorithm that knows the entire input sequence in advance.