The (co)homology of the Deligne–Mumford moduli spaces of marked rational curves

This informal note collects key results and open problems on the (co)homology of the Deligne–Mumford moduli spaces of real marked rational curves. The open problems are both of topological nature, aiming to investigate the (co)homology of these spaces further, and of algebraic nature, aiming to describe the structure of the homology of these spaces in operad-like terms.