Rapid developments in ultrafast laser technology over the past decade have enabled the generation of strong ultrashort pulses spanning the entire electromagnetic spectrum. Nonlinear excitation by such pulses allows for the manipulation of material properties on ultrafast timescales, potentially leading to new phases with novel functionalities that can be exploited for device applications. Quantum materials provide a natural playground for exploring such nonlinear phenomena as they often exhibit extreme sensitivity to small external stimuli. Of particular interest are strong pulses in the THz regime, their low photon energy can directly drive the low energy collective modes of these materials without significant heating. In this thesis, we implement such a nonlinear THz excitation scheme to study THz driven phenomena in two classes of quantum materials: superconductors and topological semimetals. Through nonlinear THz emission measurements in Nb3Sn superconducting films, we show that suitably chosen THz pulses can drive large nonlinear supercurrents resulting in symmetry-breaking effects forbidden in the equilibrium state. Further, we study the THz excitation of collective modes in the iron-based superconductor Ba(Fe0.92Co0.08)2As2. The strong interband pairing in this system results in amplitude mode behavior very different from that observed previously in conventional superconductors. Strong THz pulses can also coherently excite phonons. In the topological system ZrTe5, we demonstrate that such a phonon excitation alters the electronic structure of the material, switching its topological state on ultrafast time scales. Through these experiments, we aim to establish nonlinear THz spectroscopy as a new tool to drive materials into novel phases and exert control over their properties in ways not possible by conventional equilibrium tuning methods.