The experimental study of rare decays of hadrons containing the b quark has been a fertile ground for some time, and keeps being one of the most interesting subjects in high energy physics. It has improved our understanding of hadronic processes, and allows investigating various aspects of the Standard Model and searching for hints of physics beyond the Standard Model. Examples are the comparison of branching fractions of charmless modes with predictions of models, the constraints on CKM angles (B{sup 0} {yields} {pi}{sup +}{pi}{sup -}, B {yields} DK, with D in suppressed modes), the observation of purely leptonic modes (B{sup {+-}} {yields} {tau}{sup {+-}}{nu}), the recently established difference in A{sub CP} between B{sup 0} {yields} K{sup +}{pi}{sup -} and B{sup {+-}} {yields} K{sup {+-}}{pi}{sup 0}, suspected to be a hint new physics. All of them came from a long and successful experimental activity with e{sup +}e{sup -} collisions at the {Upsilon}(4S) resonance. With hadronic colliders now coming into play, the study of rare decays is reaching new heights. Given the high cross section for production of all kinds of B hadrons, the record luminosities now provided by the Tevatron collider, and the LHC program in view for the next years, there is the potential for a rich program of interesting new measurements, including even rarer modes as the B{sub (s)}{sup 0} {yields} {mu}{sup +}{mu}{sup -}, strongly suppressed in the standard model but very sensitive to many NP scenarios. The complexity of the hadronic collision environment, however, requires detectors with high precision and high quality tracking, and a trigger system capable of complex event selections at high rates. The CDF experiment, thanks to a fast trigger on impact parameter, has been able to reconstruct many rare B decays, including previously unobserved modes B{sub s}{sup 0} {yields} K{sup +}K{sup -} and B{sub s}{sup 0} {yields} K{sup -}{pi}{sup +}, the latter being particularly interesting for its relationship with the puzzling difference in CP asymmetry between neutral and charged modes. In this thesis we go beyond B mesons, and present the first measurements of Branching fractions and CP asymmetries in charmless b-baryon modes. We study two-body {Lambda}{sub b}{sup 0} decays into final states with a proton and a charged pion or kaon. Their branching fractions can be significantly affected by New Physics contributions; under supersymmetric models with R-parity violation, they can be increased by two orders of magnitude. Their CP-violating asymmetries are also interesting to measure in search for possible further anomalies: then may reach significant size {Omicron}(30%) in the Standard Model, and are also sensitive to possible new physics sources.
