  This article reviews how nuclear fission is described within nuclear density
functional theory. In spontaneous fission, half-lives are the main observables
and quantum tunnelling the essential concept, while in induced fission the
focus is on fragment properties and explicitly time-dependent approaches are
needed. The cornerstone of the current microscopic theory of fission is the
energy density functional formalism. Its basic tenets, including tools such as
the HFB theory, effective two-body effective nuclear potentials,
finite-temperature extensions and beyond mean-field corrections, are presented
succinctly. The EDF approach is often combined with the hypothesis that the
time-scale of the large amplitude collective motion driving the system to
fission is slow compared to typical time-scales of nucleons inside the nucleus.
In practice, this hypothesis of adiabaticity is implemented by introducing (a
few) collective variables and mapping out the many-body Schr\"odinger equation
into a collective Schr\"odinger-like equation for the nuclear wave-packet.
Scission configurations indicate where the split occurs. This collective
Schr\"odinger equation depends on an inertia tensor that includes the response
of the system to small changes in the collective variables and also plays a
special role in the determination of spontaneous fission half-lives. A
trademark of the microscopic theory of fission is the tremendous amount of
computing needed for practical applications. In particular, the successful
implementation of the theories presented in this article requires a very
precise numerical resolution of the HFB equations for large values of the
collective variables. Finally, a selection of the most recent and
representative results obtained for both spontaneous and induced fission is
presented with the goal of emphasizing the coherence of the microscopic
approaches employed.
