This article introduces Cluster Branching, a novel branching strategy for exact algorithms solving Vehicle Routing Problems (VRPs). While branching is crucial for the efficiency of branch-and-bound-based algorithms, existing branching types such as Edge Branching, CutSet Branching, and Ryan&Foster Branching have their limitations. The proposed branching strategy aggregates multiple edge variables into higher-level decision structures corresponding to clusters of customers. Through extensive computational experiments on Distance-Constrained VRP, Capacitated VRP, and VRP with Time Windows instances, we demonstrate that Cluster Branching significantly enhances the performance of a state-of-the-art Branch-Cut-and-Price algorithm, often improving branching quality and reducing the size of the search trees, which allows more instances to be solved. Beyond that, Cluster Branching is also shown to have a robust performance over different instance types, and may work well even for those with randomly positioned customers.