The radio sky is not static. Time-variable radio emission is induced in the Earth's atmosphere, and is emitted by other bodies in our Solar System, the Milky Way, and in galaxies beyond our own. It may vary on timescales which span a range from years to nanoseconds. Technological advances are increasingly allowing radio astronomers to study the radio transient events which occur on the shortest of these timescales. Of particular interest are the emission from pulsars, and an extragalactic population of fast radio bursts (FRBs), which was discovered just over a decade ago. The signals measured from both on Earth are affected by matter they traverse en-route to us. Through their dispersion measures (DMs), pulsars have enabled radio astronomers to model the Milky Way's electron distribution. Likewise, extragalactic FRBs may potentially probe matter which lies between galaxies, and that of their own hosts. To maximise the use of such distant events, the expansion of the Universe must be accounted for. The recession of other galaxies and the phenomenon of cosmological time dilation both measurably impact observed signatures of FRBs. Both effects may be accounted for by accurate deduction of FRB source's redshifts. This is driving radio astronomers to develop radio facilities with the high sensitivities and large fields of view necessary for detection of FRBs in greater numbers, and with precise enough angular resolutions to associate them with host galaxies and thus obtain their redshifts. So far only one FRB, by virtue of its observed repeating behaviour, has been localised. The e-MERLIN interferometer is a six-dish array of radio telescopes in the United Kingdom. Its long baselines between dishes provide it with the means to localise to sub-arcsecond accuracy any FRBs it may detect. This thesis details the Localisation of Fast Transients with e-MERLIN (LOFT-e) upgrade to e-MERLIN, developed by the author to enable high time-resolution studies with the instrument. The first successful observations of pulsars and single pulses from Rotating RAdio Transients (RRATs) are presented, techniques developed to increase its sensitivity via incoherent beamforming are detailed and the instrument's RFI environment is discussed. Finally, a framework to allow DM-redshift analysis of FRBs which may never be localised is introduced.