Electrical transport in materials is typically diffusive, due to dominant momentum-relaxing scattering of carriers with the phonons or defects. In ultraclean material systems such as GaAs/AlGaAs or graphene/hBN heterostructures, momentum-relaxing can be suppressed, leading to the onset of non-diffusive transport regimes, where Ohm's law is no longer valid. Within these non-diffusive regimes, the hydrodynamic regime occurs when momentum-conserving electron-electron scattering length scale is smaller than the device length scale (usually at intermediate temperatures). On the other hand, weak electron-electron scattering (at low temperatures) results in ballistic transport, commonly understood using the familiar single-particle framework of injected carriers travelling in straight line trajectories with intermittent reflections off device boundaries. Both the ballistic and hydrodynamic regimes can exhibit a emph{negative} nonlocal resistance, and collective behaviour such as the formation of current vortices. In this work, we study nonlocal current-voltage characteristics in mesoscopic devices fabricated from a GaAs/AlGaAs heterostructure that hosts a two-dimensional electron system in a GaAs quantum well. First, we report a quadratic non-linearity in the nonlocal current-voltage characteristics that manifests in any device where a nonlocal voltage measurement is possible. Using measurements at low temperatures ($sim$ 4 K) across multiple devices and considering various contact configurations for each device, we show that the non-linearity is universal. We apply the non-linearity to rectification and frequency multiplication. We also report on a periodic peaks in the nonlocal voltage vs. magnetic field, in an enclosed mesoscopic geometry in which transverse magnetic focusing (TMF) is typically studied. These peaks occur at weak magnetic fields, are independent of the source-detector separation and are distinct from TMF. Our experimental findings are backed by an extensive set of simulations using in both the semiclassical as well as quantum-coherent transport models. / Master of Science / Current is made up of charged particles such as electrons moving through a material. Typically, current is proportional to the applied voltage and flows from higher to lower potential within the device with the potential decreasing monotonically as we move from the source contact to the drain contact irrespective of the path taken through the device. This is commonly known as Ohm's law, and is followed in most materials we come across. The motion of electrons carrying this current is akin to the motion of balls inside a pinball machine, their momentum randomized by intermittent collisions due to lattice vibrations, defects and impurities present in the material. In ultraclean two-dimensional materials at low-intermediate temperatures (where lattice vibration is weak), these collisions become sparse. Collisions of electrons with other electrons now become important. When electron-electron collisions are frequent, the electrons collectively behave like a fluid, giving rise to so called hydrodynamic transport. On the other hand, when electron-electron collisions are sparse as well, electrons move unhindered in ballistic straight line trajectories until they reflect off the device boundaries. This is known as ballistic transport. Under both these transport regimes, Ohm's law breaks down, leading to interesting physical phenomena such as the formation of current whirlpools. In this work, we study the voltage measured at a point in the device which is distinct from the point where current is injected or extracted. This is commonly known as the nonlocal voltage. We explore the relationship between the nonlocal voltage and the injected current and find it to be significantly different from predictions made by Ohm's law. We use this novel current-voltage relationship to build a rectifier and frequency multiplier - two devices commonly used in high-frequency detection, radar systems and telecommunications. We also report previously unseen periodic oscillations in the nonlocal voltage when the magnetic field perpendicular to the device is varied. Using high-resolution simulations, we show the these oscillations can not be explained by looking at individual electron paths, and arise due to contribution from all electrons that travel through the device.