  In this paper, we introduce the first-order differential operators $d_0$ and
$d_1$ acting on the quaternionic version of differential forms on the flat
quaternionic space $\mathbb{H}^n$. The behavior of $d_0,d_1$ and
$\triangle=d_0d_1$ is very similar to $\partial,\overline{\partial}$ and
$\partial \overline{\partial}$ in several complex variables. The quaternionic
Monge-Amp\`{e}re operator can be defined as $(\triangle u)^n$ and has a simple
explicit expression. We define the notion of closed positive currents in the
quaternionic case, and extend several results in complex pluripotential theory
to the quaternionic case: define the Lelong number for closed positive
currents, obtain the quaternionic version of Lelong-Jensen type formula, and
generalize Bedford-Taylor theory, i.e., extend the definition of the
quaternionic Monge-Amp\`{e}re operator to locally bounded quaternionic
plurisubharmonic functions and prove the corresponding convergence theorem.
