  We formulate Barrow holographic dark energy, by applying the usual
holographic principle at a cosmological framework, but using the Barrow entropy
instead of the standard Bekenstein-Hawking one. The former is an extended
black-hole entropy that arises due to quantum-gravitational effects which
deform the black-hole surface by giving it an intricate, fractal form. We
extract a simple differential equation for the evolution of the dark energy
density parameter, which possesses standard holographic dark energy as a
limiting sub-case, and we show that the scenario can describe the universe
thermal history, with the sequence of matter and dark energy eras.
Additionally, the new Barrow exponent $\Delta$ significantly affects the dark
energy equation of state, and according to its value it can lead it to lie in
the quintessence regime, in the phantom regime, or experience the
phantom-divide crossing during the evolution.
