A big classification result that I'm aware of is due to Gromov and Lawson.

Theorem. Let $M$ be a compact simply connected manifold of dimension $\geq 5$, which is not a spin. Then $M$ admits a metric of positive scalar curvature.

A big classification result that I'm aware of is due to Gromov and Lawson.

Theorem. Let $M$ be a compact simply connected manifold of dimension $\geq 5$, which is not a spin. Then $M$ admits a metric of positive scalar curvature. A spin manifold of dimension $\geq 5$ carries a metric of positive scalar curvature iff its $\alpha$-genus is zero.

The Seiberg-Witten invariants provide special obstructions to existence of a metric of positive scalar curvature in dimension 4.

There are two good survey articles on the subject by J. Rosenberg (link 1) and S. Stolz (link 2).

Here are two big classification results that I'm aware of.

Theorem 1 (Gromov-Lawson). Let $M$ be a compact simply connected manifold of dimension $\geq 5$, which is not a spin. Then $M$ admits a metric of positive scalar curvature.

Theorem 2 (Kazdan-Warner). Every manifold carries a metric of constant negative scalar curvature.

A big classification result that I'm aware of is due to Gromov and Lawson.

Theorem. Let $M$ be a compact simply connected manifold of dimension $\geq 5$, which is not a spin. Then $M$ admits a metric of positive scalar curvature.