Efficient simulation of quantum systems is one of the fundamental problems of quantum physics and chemistry. The computational costs of all known exact methods for quantum simulation using classical computers grow exponentially with system size. Here, we demonstrate that quantum computation could provide exact, polynomial-time simulation of chemical reactions. In particular, we show that a quantum computer with one hundred qubits could simulate the complete quantum dynamics of a lithium atom, a task which is impossible on current classical computers. In contrast to the classical case, we demonstrate that for most chemical reactions it is more efficient to perform the quantum simulation of the full electronic dynamics than to use potential energy surfaces in the Born-Oppenheimer approximation. Furthermore, we demonstrate how to efficiently obtain observables of chemical interest, such as state-to-state transition probabilities and reaction rates.