  Many aspects of pluripotential theory are generalized to quaternionic
$m$-subharmonic functions. We introduce quaternionic version of notions of the
$m$-Hessian operator, $m$-subharmonic functions, $m$-Hessian measure,
$m$-capapcity, the relative $m$-extremal function and the $m$-Lelong number,
and show various propositions for them, based on $d_0$ and $ d_1$ operators,
the quaternionic counterpart of $\partial$ and $\overline{\partial}$, and
quaternionic closed positve currents. The definition of quaternionic
$m$-Hessian operator can be extended to locally bounded quaternionic
$m$-subharmonic functions and the corresponding convergence theorem is proved.
The comparison principle and the quasicontinuity of bounded quaternionic
$m$-subharmonic functions are established. We also find the fundamental
solution of the quaternionic $m$-Hessian operator.
