The high degree of controllability in modern day quantum simulators has enabled the engineering of Floquet topological Bloch bands and even the realization of novel non-equilibrium states like the discrete time crystal, through the implementation of time-dependent driving. Such drives, though, have so far been largely limited to time-periodic (i.e., Floquet) modulations, and their time-quasiperiodic (i.e., multi-tone) generalizations. This leads to a natural question: what are more general classes of quantum drives, conceptually distinct from Floquet or time-quasiperiodic, which lead to interesting phase structure in dynamics? In this talk, I will present a general theoretical construction of quantum drives in which the position of a classical particle moving autonomously on a smooth connected manifold is used to steer a quantum Hamiltonian over time, thus leading to myriad time-dependent quantum Hamiltonians with properties dependent on the choice of underlying manifold and trajectory. Applying the construction we dub "geometric quantum driving" to a compact 2d hyperbolic Riemann surface, I will demonstrate a new class of drives called "hyperbolically driven quantum systems", and further show how in the adiabatic limit they are topologically classified by a quantized response that can be extracted through the dynamics of a simple local observable. We propose geometric quantum driving to be a general framework to chart the landscape of time-dependent quantum systems and investigate the universal phase structures they exhibit, as well as a useful tool to enhance the capabilities of modern day quantum simulators.