  Polycrystalline samples of four intermetallic compounds: MV$_2$Al$_{20}$ (M =
Sc, Y, La, and Lu) were synthesized using an arc-melting technique. The crystal
structures were analyzed by means of powder x-ray diffraction and Rietveld
analysis, and the physical properties were studied by means of heat capacity,
electrical resistivity and magnetic susceptibility measurements down to 0.4 K.
For ScV$_2$Al$_{20}$, LuV$_2$Al$_{20}$, and YV$_2$Al$_{20}$, superconductivity
was observed with critical temperatures T$_c$ = 1.00 K, 0.57 K, and 0.60 K,
respectively. Superconductivity for the Lu compound is reported for the first
time. Theoretical calculations of the electronic and phonon structures were
conducted in order to analyze the superconductivity and dynamics in
ScV$_2$Al$_{20}$, YV$_2$Al$_{20}$, and LuV$_2$Al$_{20}$ and to explain the lack
of a superconducting transition in LaV$_2$Al$_{20}$ down to 0.4 K. The results
of the experimental and theoretical studies show that all the compounds are
weakly-coupled type II BCS superconductors, and reveal the importance of the
M-atom anharmonic "rattling" modes for the superconductivity in these
materials, which seem to enhance T$_c$, especially for ScV$_2$Al$_{20}$.
