Algebraic (semi-) Riemannian geometry ? I hope these are not to vague questions for MO. 
Is there an analog of the concept of a Riemannian metric, in algebraic geometry? 
Of course, transporting things literally from the differential geometric context, we have to forget about the notion of positive definiteness, cause a bare field has no ordering. So perhaps we're looking to an algebro geometric analog of semi- Riemannian geometry.
Suppose to consider a pair $(X,g)$, where $X$ is a (perhaps smooth) variety and $g$ is a nondegenerate section  of the second symmetric power of the tangent bundle (or sheaf) of $X$.
What can be said about this structure? Can some results of DG be reproduced in this context? Is there a literature about this things?
 A: Joel Kamnitzer had a very similar question a couple years ago, that prompted a nice discussion at the Secret Blogging Seminar.  I'm afraid no one ended up citing any literature, and I have been unable to find anything with a quick Google search, but that doesn't rule out the possibility of existence.
A: This topic in the affine case is extensively studied in Ernst Kunz unpublished book "Algebraic Differential Calculus". You can get it as a collection of several PS files at his webpage (scroll to the bottom):
Kunz' webpage
A: If a holomorphic Riemannian metric $g=g_{ij}(z) dz^i dz^j$ on a compact Kaehler manifold $X$ is everywhere nondegenerate, then the metric has a holomorphic Levi-Civita connection, so the Atiyah class of the tangent bundle of $X$ is zero. Therefore a finite etale cover of $X$ is a complex torus, and the metric pulls back to be translation invariant. Hence $X$ is a quotient of such a torus by a finite group acting as affine isometries without fixed points. On the other hand, if you allow degeneracies of the holomorphic Riemannian metric, I suppose anything could happen. If you allow $X$ to be a compact complex manifold, perhaps not Kaehler, then you might look at the papers of Sorin Dumitrescu where you find a low dimensional classification.
A: It is also rather natural to look at holomorphic conformal structure given locally by holomorphic Riemannian metrics up to multiplication by invertible functions. More precisely,
a holomorphic conformal structure is given by a  nowhere degenerate section $\omega \in H^0(X,Sym^2\Omega^1_X \otimes \mathcal L)$ where $\mathcal L$ is a line-bundle.
The classification of holomorphic conformal structures on compact complex surfaces is carried out by Kobayashi and Ochiai in  Holomorphic structures modeled after hyperquadratics, Tohoku Math. J. 34, 587-629 (1982).
There is also a classification in the case of projective $3$-folds in this paper.
