The time-variable long-wavelength sky harbours a number of known but unsolved astrophysical problems, and surely many more undiscovered phenomena. With modern tools such problems will become tractable, and new classes of astronomical objects will be revealed. These tools include digital telescopes made from powerful computing clusters, and improved theoretical methods. In this thesis we employ such devices to understand better several puzzles in the time-domain radio sky. Our primary focus is on the origin of fast radio bursts (FRBs), a new class of transients of which there seem to be thousands per sky per day. We offer a model in which FRBs are extragalactic but non-cosmological pulsars in young supernova remnants. Since this theoretical work was done, observations have corroborated the picture of FRBs as young rotating neutron stars, including the non-Poissonian repetition of FRB 121102. We also present statistical arguments regarding the nature and location of FRBs. These include reinstituting the classic V/Vmax-test to measure the brightness distribution of FRBs, i.e., constraining dlogN/dlogS. We find consistency with a Euclidean distribution. This means current observations cannot distinguish between a cosmological population and a more local uniform population, unless added assumptions are made. We also showed that the rate of FRBs at low frequencies is consistent with the rate at 1.4 GHz, which is promising for upcoming high-impact experiments. One of these is the Canadian Hydrogen Intensity Mapping Experiment (CHIME). We outline this instrument and its three back-ends: a cosmology experiment whose goal is to measure dark energy through 21 cm intensity mapping, a pulsar back-end, and an FRB project that is expected to be by far the fastest survey in the foreseeable future. We describe the creation of a digital beamforming back-end on the CHIME Pathfinder, which acts as a test-bed for the three final experiments just described. We also discuss the commissioning of a 24/7 real-time VLBI FRB search between the Pathfinder's synthetic beam and the Algonquin Radio Observatory (ARO) 46m telescope, including early results. Finally, we present a study of the microstructure in B0329+54's individual pulses in full-polarization and present results on its quasi-periodic structure.