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In this PhD thesis, we develop a new approach to classical gravity startingfrom Mach's principles and the idea that the local shape of spatialconfigurations is fundamental. This new theory, "shape dynamics", is equivalentto general relativity but differs in an important respect: shape dynamics is atheory of dynamic conformal 3-geometry, not a theory of spacetime. Equivalenceis achieved by trading foliation invariance for local conformal invariance (upto a global scale). After the trading, what is left is a gauge theory invariantunder 3d diffeomorphisms and conformal transformations that preserve the volumeof space. The local canonical constraints are linear and the constraint algebracloses with structure constants. Shape dynamics, thus, provides a novel newstarting point for quantum gravity.  The procedure for the trading of symmetries was inspired by a techniquecalled "best matching". We explain best matching and its relation to Mach'sprinciples. The key features of best matching are illustrated through finitedimensional toy models. A general picture is then established where relationaltheories are treated as gauge theories on configuration space. Shape dynamicsis then constructed by applying best matching to conformal geometry. We thenstudy shape dynamics in more detail by computing its Hamiltonian andHamilton-Jacobi functional perturbatively.  This thesis is intended as a pedagogical but complete introduction to shapedynamics and the Machian ideas that led to its discovery. The reader isencouraged to start with the introduction, which gives a conceptual outline andlinks to the relevant sections in the text for a more rigorous exposition. Whenfull rigor is lacking, references to the literature are given. It is hoped thatthis thesis may provide a starting point for anyone interested in learningabout shape dynamics.

Shape dynamics and Mach's principles: Gravity from conformal geometrodynamics