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In this thesis we build a geometric theory of Hamiltonian Monte Carlo, with an emphasis on symmetries and its bracket generalisations, construct the canonical geometry of smooth measures and Stein operators, and derive the complete recipe of measure-constraints preserving dynamics and diffusions on arbitrary manifolds. Specifically, we will explain the central role played by mechanics with symmetries to obtain efficient numerical integrators, and provide a general method to construct explicit integrators for HMC on geodesic orbit manifolds via symplectic reduction. Following ideas developed by Maxwell, Volterra, Poincaré, de Rham, Koszul, Dufour, Weinstein, and others, we will then show that any smooth distribution generates considerable geometric content, including ''musical" isomorphisms between multi-vector fields and twisted differential forms, and a boundary operator - the rotationnel, which, in particular, engenders the canonical Stein operator. We then introduce the ''bracket formalism" and its induced mechanics, a generalisation of Poisson mechanics and gradient flows that provides a general mechanism to associate unnormalised probability densities to flows depending on the score pointwise. Most importantly, we will characterise all measure-constraints preserving flows on arbitrary manifolds, showing the intimate relation between measure-preserving Nambu mechanics and closed twisted forms. Our results are canonical. As a special case we obtain the characterisation of measure-preserving bracket mechanical systems and measure-preserving diffusions, thus explaining and extending to manifolds the complete recipe of SGMCMC. We will discuss the geometry of Stein operators and extend the density approach by showing these are simply a reformulation of the exterior derivative on twisted forms satisfying Stokes' theorem. Combining the canonical Stein operator with brackets allows us to naturally recover the Riemannian and diffusion Stein operators as special cases. Finally, we shall introduce the minimum Stein discrepancy estimators, which provide a unifying perspective of parameter inference based on score matching, contrastive divergence, and minimum probability flow.