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Since the time of Darwin, scientists have struggled to reconcile the evolution of biological forms in a universe determined by fixed laws. These laws underpin the origin of life, evolution, human culture and technology, as set by the boundary conditions of the universe, however these laws cannot predict the emergence of these things. By contrast evolutionary theory works in the opposite direction, indicating how selection can explain why some things exist and not others. To understand how open-ended forms can emerge in a forward-process from physics that does not include their design, a new approach to understand the non-biological to biological transition is necessary. Herein, we present a new theory, Assembly Theory (AT), which explains and quantifies the emergence of selection and evolution. In AT, the complexity of an individual observable object is measured by its Assembly Index (a), defined as the minimal number of steps needed to construct the object from basic building blocks. Combining a with the copy number defines a new quantity called Assembly which quantifies the amount of selection required to produce a given ensemble of objects. We investigate the internal structure and properties of assembly space and quantify the dynamics of undirected exploratory processes as compared to the directed processes that emerge from selection. The implementation of assembly theory allows the emergence of selection in physical systems to be quantified at any scale as the transition from undirected-discovery dynamics to a selected process within the assembly space. This yields a mechanism for the onset of selection and evolution and a formal approach to defining life. Because the assembly of an object is easily calculable and measurable it is possible to quantify a lower limit on the amount of selection and memory required to produce complexity uniquely linked to biology in the universe.