Studies of $C\!P$-violation in charmless three-body $b$-hadron decays

Violation of combined charge and parity inversion ($C\!P$) is a property of the Standard Model that results in a fundamental difference between particles and anti-particles. The single source of $C\!P$-violation in the Standard Model is insufficient to explain the dominance of matter over anti-matter in the contemporary universe, however, thus far, there has been no clear observation of $C\!P$-violation beyond the Standard Model. Constraints on various $C\!P$-violating observables are now precise enough that these represent sensitive tests for physics beyond the Standard Model. This thesis firstly documents the observation of two three-body $b$-baryon decays, and measurements of their phase-space integrated $C\!P$-asymmetries, which are some of the first to be performed on baryon decays. These measurements provide useful information on hadronisation in $b$-baryon decays, on the intermediate decay dynamics, and give a potential avenue to search for $C\!P$-violation in baryon decays. An amplitude analysis of the $\B^+ \rightarrow \pi^+\pi^+\pi^-$ decay is also performed, where sizable $C\!P$-violation was observed previously, and the $C\!P$-violating parameters that relate to the intermediate resonant structure extracted. This is achieved by formulating a model for the decay amplitude, which is dominated by the $\rho(770)^0$ resonance and various broad overlapping scalar resonances. The scalar contributions to this decay are modelled using the unitarity-conserving K-matrix model, which relies on historical scattering data for the relative couplings between the scalar resonances and between the possible final states. These results provide insight into the mechanism by which the $C\!P$-violation in the Standard Model manifests in practice, give information on SU(3)-flavour symmetry breaking phenomena in hadron decays, and are a valuable input into future measurements of the CKM unitarity triangle angles.