Rapid urbanisation and population growth is driving unprecedented levels of building construction. Over the next 40 years, approximately 230 billion square meters of new floor area will be constructed globally, a doubling of existing building stock. Already, the production of concrete and steel accounts for a third of worldwide industrial CO2 emissions, representing a major opportunity, and responsibility, for structural engineers to contribute towards a low-carbon future through efficient design. A significant majority of the structural material in a typical building exists within the floors, making these a prime target for material reductions. This dissertation shows that thin shell concrete floors are a viable alternative to typical slabs and beams in multi-storey buildings. Switching the dominant structural behaviour from bending to membrane action increases efficiency, enabling significant embodied carbon reductions. A system is proposed featuring pre-cast textile reinforced concrete shells of uniform thickness and shallow depth, supported at columns, with a network of prestressed steel tension ties. A lightweight foamed concrete fill is cast above the shells to provide a level top surface and transfer floor loads to the shell. The structural behaviour of this system is explored through a series of computational and experimental investigations, leading to refinement of the design, exploration of construction methods and the development of a complete design methodology incorporating novel theoretical work. The shells feature optimised singly-curved groin vault geometry. This provides efficient structural performance whilst simultaneously minimising construction complexity. Thus, a practical and scalable solution is proposed, which is shown to offer considerable embodied carbon savings over typical concrete and steel floor structures. This work provides a robust platform for future refinement and large-scale implementation of thin-shell concrete floors for sustainable buildings.