The focus of this thesis is the study of superluminous supernovae (SLSNe) within the Pan- STARRSI Medium Deep Survey (PSI MDS). Previous work on SLSNe has displayed a trend in their host environments, in that they seem to preferentially occur in low luminosity or dwarf galaxies. This presented the motivation to search for supernovae in faint host galaxies in the hope of finding new SLSNe. The PS 1 MDS offered an ideal experiment for such a search, in the redshift range z = 0.3 - 1.5. During the 3 year period of study, approximately 400 PSI MDS hostless transients were catalogued and around half classified. Of the 22 spectroscopically confirmed supernovae which were not of Type la, over 50% have been classified as likely SLSNe. This led to the discovery and detailed study of PS1-11ap (in a faint host galaxy at z = 0.524). The object was discovered at the start of an observing season resulting in a superbly sampled light curve. Rest frame UV and optical spectra were obtained throughout its observing season, including a host spectrum. From comparisons with known SLSNe and modelling of a composite bolometric light curve a confident classification of PS 1-11ap as a rare, slowly evolving SLSNe-Ic is made. PSI-lOpm and PSI-I0ahf, at z = 1.206 and z = 1.158, were also discovered. Spectroscopic and photometric comparisons of PSI-lOpm with SLSNe-Ic place it comfortably within this class . PSI-lOahf sits better as a slowly evolving SLSN-Ic, however alternative progenitor possibilities are presented. A single spectrum and appropriate photometric magnitude for PS1-11acn (z = 0.61) offer limited evidence for SLSN typing. Rates for the two SLSNe-Ic classes are also estimated using Monte-Carlo simulations, at around 10"-4 and 10"-5 that of the normal CCSNe rate.