This thesis is devoted to the study of extremely energetic short-timescale astrophysical events, Gamma-ray bursts (GRBs). GRBs exhibit broad-band bright non-thermal emission, which was analysed using two major experiments: the High Altitude Water Cherenkov observatory (HAWC) and the High Energy Stereoscopic System (H.E.S.S.). The two experiments are in many respects complementary for the observation of very high energy (VHE) gamma-ray emission from GRBs, and in this work the respective advantages were exploited to maximise the sensitivity to VHE signals. After the analysis of several tens of GRBs observed using H.E.S.S. until 2017, where no significant emission was detected, improvements in the observation strategy of H.E.S.S. allowed the detection of GRB 180729B and GRB 190829A. These detections are presented in context with multi-wavelength data, proposing plausible emission mechanisms, thus concluding a decade-long search for these elusive phenomena at VHE. In the second part, novel methods to improve the accuracy of the HAWC detector simulation are presented, including better modelling of the detector efficiencies and electronics. A model that accounts for the detector response and the GRB flux evolution has been developed to estimate the optimal integration time for VHE searches with HAWC. Thanks to these improvements, it is possible to exploit the wide field of view and high duty cycle of HAWC for the search of VHE emission in several tens of GRBs. Preliminary evidence for emission is found in one of the GRBs studied, and upper limits are obtained for all the GRBs analysed and placed in context of the X-ray properties of these events. Finally, the limits and detections presented in this work are placed within the framework of the current understanding of GRBs and prospects for future and present VHE gamma-ray detectors are presented
