Cluster Editing, also known as Correlation Clustering, is a well-studied graph modification problem. In this problem, one is given a graph and the task is to perform up to $k$ edge additions or deletions to transform it into a cluster graph, i.e., a graph consisting of a disjoint union of cliques. However, in real-world networks, clusters are often overlapping. For example in social networks, a person might belong to several communities - e.g. those corresponding to work, school, or neighborhood. Other strong motivations come from biological network analysis and from language networks. Trying to cluster words with similar usage in the latter can be confounded by homonyms, that is, words with multiple meanings like "bat." In this paper, we introduce a new variant of Cluster Editing whereby a vertex can be split into two or more vertices. First used in the context of graph drawing, this operation allows a vertex $v$ to be replaced by two vertices whose combined neighborhood is the neighborhood of $v$ (and thus $v$ can belong to more than one cluster). We call the new problem Cluster Editing with Vertex Splitting and we initiate the study of it. We show that it is NP-complete and fixed-parameter tractable when parameterized by the total number $k$ of allowed vertex-splitting and edge-editing operations. In particular, we obtain an $O(2^{9k log k} + n + m)$-time algorithm and a $6k$-vertex kernel.