Global properties of biconservative surfaces in ${\mathbb{R}}^3$ and ${\mathbb{S}}^3$

We survey some recent results on biconservative surfaces in $3$-dimensional space forms $N^3(c)$ with a special emphasis on the $c=0$ and $c=1$ cases. We study the local and global properties of such surfaces, from extrinsic and intrinsic point of view. We obtain all non-$CMC$ complete biconservative surfaces in ${\mathbb{R}}^3$ and ${\mathbb{S}}^3$.