Given a complex three-dimensional domain delimited by a closed and non-degenerate input triangle mesh without any self-intersection, a common geometry processing task consists in cutting up the domain into cells through a set of planar cuts, creating a "cut-cell mesh", i.e., a volumetric decomposition of the domain amenable to visualization (e.g., exploded views), animation (e.g., virtual surgery), or simulation (finite volume computations). A large number of methods have proposed either efficient or robust solutions, sometimes restricting the cuts to form a regular or adaptive grid for simplicity; yet, none can guarantee both properties, severely limiting their usefulness in practice. At the core of the difficulty is the determination of topological relationships among large numbers of vertices, edges, faces and cells in order to assemble a proper cut-cell mesh: while exact geometric computations provide a robust solution to this issue, their high computational cost has prompted a number of faster solutions based on, e.g., local floating-point angle sorting to significantly accelerate the process --- but losing robustness in doing so. In this paper, we introduce a new approach to planar cutting of 3D domains that substitutes topological inference for numerical ordering through a novel mesh data structure, and revert to exact numerical evaluations only in the few rare cases where it is strictly necessary. We show that our novel concept of topological cuts exploits the inherent structure of cut-cell mesh generation to save computational time while still guaranteeing exactness for, and robustness to, arbitrary cuts and surface geometry. We demonstrate the superiority of our approach over state-of-the-art methods on almost 10,000 meshes with a wide range of geometric and topological complexity. We also provide an open source implementation.