  In arXiv:1802.02833 Guichard and Wienhard introduced the notion of
$\Theta$-positivity, a generalization of Lusztig's total positivity to real Lie
groups that are not necessarily split. Based on this notion, we introduce in
this paper $\Theta$-positive representations of surface groups. We prove that
$\Theta$-positive representations are $\Theta$-Anosov. This implies that
$\Theta$-positive representations are discrete and faithful and that the set of
$\Theta$-positive representations is open in the representation variety. We
show that the set of $\Theta$-positive representations is closed within the set
of representations that do not virtually factor through a parabolic subgroup.
From this we deduce that for any simple Lie group $\mathsf G$ admitting a
$\Theta$-positive structure there exist components consisting of
$\Theta$-positive representations. More precisely we prove that the components
parametrized using Higgs bundles methods in arXiv:2101.09377 consist of
$\Theta$-positive representations.
