  This article reviews several definitions of the fractional Laplace operator
(-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional
derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of
continuous functions vanishing at infinity and on the space C_{bu} of bounded
uniformly continuous functions. Among these definitions are ones involving
singular integrals, semigroups of operators, Bochner's subordination and
harmonic extensions. We collect and extend known results in order to prove that
all these definitions agree: on each of the function spaces considered, the
corresponding operators have common domain and they coincide on that common
domain.
