The prospect of cooling matter down to temperatures that are close to the absolute zero raises intriguing questions about how chemical reactivity changes under these extreme conditions. Although some types of chemical reaction still occur at 1 µK, they can no longer adhere to the conventional picture of reactants passing over an activation energy barrier to become products. Indeed at ultracold temperatures, the system enters a fully quantum regime, and quantum mechanics replaces the classical picture of colliding particles. In this Review, we discuss recent experimental and theoretical developments that allow us to explore chemical reactions at temperatures that range from 10 K to 500 nK. Although the field is still in its infancy, exceptional control has already been demonstrated over reactivity at low temperatures.ultracold temperatures, the classical description of the collision of particles needs to be replaced with a quantum description and the picture of a ball rolling over a hill needs to be replaced by that of a wave propagating across a barrier.The quantum regime is a physical regime in which our basic understanding of how chemical processes happen, in terms of molecules bumping into each other and breaking and remaking bonds, needs to be challenged. Strictly speaking, the dynamical system of moving and colliding molecules needs to be described as a superposition of partial waves, the components of which represent the different angular momentum states associated with the relative motion of the colliding particles. As the temperature is lowered, the system moves towards a regime in which it can be described by just a few quantised collisional angular momentum states (and, ultimately, just a single state). We denote the collisions as 's-wave' or 'p-wave' collisions (known as the partial-wave description) to represent those with zero or one unit of collisional angular momentum, respectively. Furthermore, in the quantum regime the population of molecular quantum states is distributed over just a few rovibrational states and ultimately collapses to a single quantum state at the lowest temperatures (under conditions of thermal equilibrium). At low temperatures, quantum effects on the rate coefficient and other properties of the reaction, such as collision energy resonances, quantum reflection and geometric phase (described in detail in the 'Reactions in the ultracold regime' section), are most likely to be observable. This is in contrast to what we observe at room temperature, where we understand the overall rate of a process to be an average of the behaviours of individual collisions of molecules with vastly varying sets of quantum numbers and a distribution of energies.When the temperature reaches sub-µK, the translational wavelength typically exceeds the average separation of molecules in the sample, even for a low density gas, and ultimately the system may enter the regime of quantum degeneracy (for example, bosonic particles that form a Bose-Einstein condensate, in which all particles are in the same quan...