Based on the differential conformal transformation in the fractional order, we defined the fractional memristor in contrast to the traditional (integer-order) memristor. As an example, a typical spin-transfer torque (STT) memristor (with the asymmetric resistance hysteresis) was proved to be a 0.8 fractional memristor. In conclusion, many memristors should not be treated as ideal ones due to the fractional interaction between flux and charge. Indeed, unless a non-ideal memristor is properly modelled as a fractional memristor, no deep physical understanding would be possible to develop a reliable commercial product.