  In order to apply holography and entropy relations to the whole universe,
which is a gravitational and thus nonextensive system, for consistency one
should use the generalized definition for the universe horizon entropy, namely
Tsallis nonextensive entropy. We formulate Tsallis holographic dark energy,
which is a generalization of standard holographic dark energy quantified by a
new dimensionless parameter $\delta$, possessing the latter as a particular
sub-case. We provide a simple differential equation for the dark energy density
parameter, as well as an analytical expression for its equation-of-state
parameter. In this scenario the universe exhibits the usual thermal history,
namely the successive sequence of matter and dark-energy epochs, before
resulting in a complete dark energy domination in the far future. Additionally,
the dark energy equation-of-state parameter presents a rich behavior and,
according to the value of $\delta$, it can be quintessence-like, phantom-like,
or experience the phantom-divide crossing before or after the present time.
Finally, we confront the scenario with Supernovae type Ia and Hubble parameter
observational data, and we show that the agreement is very good, with $\delta$
preferring a value slightly larger than its standard value 1.
