  Elastic constants and their derived properties of various cubic Heusler
compounds were calculated using first-principles density functional theory. To
begin with, Cu$_2$MnAl is used as a case study to explain the interpretation of
the basic quantities and compare them with experiments. The main part of the
work focuses on Co$_2$-based compounds that are Co$_2$Mn$M$ with the main group
elements $M=$~Al, Ga, In, Si, Ge, Sn, Pb, Sb, Bi, and Co$_2TM$ with the main
group elements Si or Ge, and the $3d$ transition metals $T=$~Sc, Ti, V, Cr, Mn,
and Fe. It is found that many properties of Heusler compounds correlate to the
mass or nuclear charge $Z$ of the main group element.
  Blackman's and Every's diagrams are used to compare the elastic properties of
the materials, whereas Pugh's and Poisson's ratios are used to analyze the
relationship between interatomic bonding and physical properties. It is found
that the {\it Pugh's criterion} on brittleness needs to be revised whereas {\it
Christensen's criterion} describes the ductile--brittle transition of Heusler
compounds very well. The calculated elastic properties give hint on a metallic
bonding with an intermediate brittleness for the studied Heusler compounds.
  The universal anisotropy of the stable compounds has values in the range of
$0.57 <A_U <2.73$. The compounds with higher $A_U$ values are found close to
the middle of the transition metal series. In particular, Co$_2$ScAl with
$A_U=0.01$ is predicted to be an isotropic material that comes closest to an
ideal Cauchy solid as compared to the remaining Co$_2$-based compounds. Apart
from the elastic constants and moduli, the sound velocities, Debye
temperatures, and hardness are predicted and discussed for the studied systems.
The calculated slowness surfaces for sound waves reflect the degree of
anisotropy of the compounds.
