  This paper, that will appear in the Notices of the AMS, begins with a brief
historical account of the beginnings of fractional calculus and the crucial
roles played by Leibniz and Fourier. Fourier's definition of fractional
derivative is introduced and unpacked by using the modern technique known as
the method of semigroups. Recent advances on the theory of fractional
derivatives are presented. Furthermore, we address some questions that have
been raised by some in the scientific community. Finally, we present three
different applications: population growth with memory, viscoleastic materials
and anomalous diffusions.
