For a smooth projective variety $X$ over $\mathbb{C}$, the Deligne cohomology groups $H_{\mathcal{D}}^p(X;\mathbb{Z}(q))$ form an important set of bigraded invariants. It comes equipped
with maps to the singular cohomology $H_{\mathcal{D}}^p(X;\mathbb{Z}(q)) \to H_{\text{sing}}^p(X(\mathbb{C}); (2\pi i)^q\mathbb{Z})$.
In a recent work with Paulo Lima-Filho, we have introduced a version of Deligne cohomology with integer coefficients for varieties defined over $\mathbb{R}$. In this theory
the role of singular cohomology is played by equivariant cohomology (\emph{w.r.t.} $\mathbb{Z}/2=\text{Gal}(\mathbb{C}/\mathbb{R})$), which is a bigaded cohomology theory. In this talk,
we will briefly overview the main properties of these two sets of bigraded invariants for real varieties and describe the ring structures they give
for projective smooth real curves.
