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The results of two separate searches for the rare two-body charmless baryonic decays B0 -> p pbar and B0s -> p pbar at the LHCb experiment are reported in this thesis. The first analysis uses a data sample, corresponding to an integrated luminosity of 0.9 fb^-1, of proton-proton collision data collected by the LHCb experiment at a centre-of-mass energy of 7 TeV. An excess of B0 -> p pbar candidates with respect to background expectations is seen with a statistical significance of 3.3 standard deviations. This constitutes the first evidence for a two-body charmless baryonic B0 decay. No significant B0s -> p pbar signal was observed. However, a small excess of B0s -> p pbar events allowed the extraction of two sided confidence level intervals for the B0s -> p pbar branching fraction using the Feldman-Cousins frequentist method. This improved the upper limit on the B0s -> p pbar branching fraction by three orders of magnitude over previous bounds. The 68.3% confidence level intervals on the branching fractions were measured to be BF(B0 -> p pbar) = ( 1.47 ^{+0.62}_{-0.51} ^{+0.35}_{-0.14} ) x 10^-8, BF(B0s -> p pbar) = ( 2.84 ^{+2.03}_{-1.68} ^{+0.85}_{-0.18} ) x 10^-8, where the first uncertainty is statistical and the second is systematic. The second analysis followed on from the first LHCb result and included the full 2011 and 2012 samples of proton-proton collision data at centre of mass energies of 7 and 8 TeV, corresponding to a total integrated luminosity of 3.122 fb^-1.