Summary: In this paper, using the concept and tools of fractional calculus, we introduce a definition for `fractional-order' multipoles of electric-charge densities, and we show that as far as their scalar potential distributions are concerned, such fractional-order multipoles effectively behave as `intermediate' sources bridging the gap between the cases of integer-order point multipoles such as point monopoles, point dipoles, point quadrupoles, etc. This technique, which involves fractional differentiation or integration of the Dirac delta function, provides a tool for formulating an electric source distribution whose potential functions can be obtained by using fractional differentiation or integration of potentials of integer-order point-multipoles of lower or higher orders. As illustrative examples, the cases of three-dimensional (point source) and two-dimensional (line source) problems in electrostatics are treated in detail, and the extension to the time-harmonic case is also addressed. In the three-dimensional electrostatic example, we suggest an electric-charge distribution which can be regarded as an `intermediate' case between cases of the electric-point monopole (point charge) and the electric-point dipole (point dipole), and we present its electrostatic potential, which behaves as \(r^{-(1+\alpha)}P_\alpha (-\cos \theta)\), where \(0<\alpha <1\) and \(P_\alpha(\cdot)\) is the Legendre function of noninteger degree \(\alpha\), thus denoting this charge distribution as a fractional \(2^\alpha\)-pole. At the two limiting cases of \(\alpha=0\) and \(\alpha=1\), this fractional \(2^\alpha\)-pole becomes the standard point monopole and point dipole, respectively. A corresponding intermediate fractional-order multipole is also given for the two-dimensional electrostatic case. Potential applications of this treatment to the image method in electrostatic problems are briefly mentioned. Physical insights and interpretation for such fractional-order \(2^\alpha\)-poles are also given.