Despite decades of study, we still do not fully understand why some massive galaxies abruptly switch off their star formation in the early Universe, and what causes their rapid transition to the red sequence. Post-starburst galaxies (PSBs) are key to understanding this transition phase, as they represent systems in which a major burst of star formation was rapidly quenched within the last Gyr. Recently, a new photometric PCA technique, has identified over 900 candidate PSBs in the UDS field at redshifts 0.5 to 2. In this conference poster, I present spectroscopic verification of this technique and demonstrate that ~80% of the photometrically-selected PSB candidates show spectral signatures characteristic of this population. I also present results on the morphological structure of these galaxies, obtained from a combination of deep ground- and space-based imaging (UDS UKIRT-WFCAM and CANDELS HST-ACS/WFC3). Using both 2D-Sersic modelling and an independent isophotal analysis, I show that PSB galaxies at high redshift (z above 1) are surprisingly compact and spheroidal, while at lower redshifts they are more extended and disc-dominated. I also discuss what these results can tell us about the potential quenching mechanisms operating in this important transitional population at different epochs.