  Let $X$ be a smooth projective curve of genus $g$, defined over an
algebraically closed field $k$, and let $G$ be a connected reductive group over
$k$. We say that a $G$-torsor is essentially finite if it admits a reduction to
a finite group, generalising the notion of essentially finite vector bundles to
arbitrary groups $G$. We give a Tannakian interpretation of such torsors, and
we prove that all essentially finite $G$-torsors have torsion degree, and that
the degree is 0 if $X$ is an elliptic curve. We then study the density of the
set of $k$-points of essentially finite $G$-torsors of degree $0$, denoted
$M_{G}^{\text{ef},0}$, inside $M_{G}^{\text{ss},0}$, the $k$-points of all
semistable degree 0 $G$-torsors. We show that when $g=1$,
$M_{G}^{\text{ef}}\subset M_{G}^{\text{ss},0}$ is dense. When $g>1$ and when
$\text{char}(k)=0$, we show that for any reductive group of semisimple rank 1,
$M_{G}^{\text{ef},0}\subset M_{G}^{\text{ss},0}$ is not dense.
