Honeycombs and Densities, including Multiple Bubbles in Gauss Space

The $n$-bubble problem seeks the least-perimeter way to enclose and separate $n$ prescribed volumes in $\mathbb{R}^m$. The solution is known only for $n = 1$ or 2 in $\mathbb{R}^m$ (round sphere and standard double bubble) and $n = 3$ in $\mathbb{R}^2$ (standard triple bubble). If you give $\mathbb{R}^m$ Gaussian density, the solution was recently proved by Milman and Neeman for $n \le m$. There is further news for other densities.
In 2000 Hales proved that regular hexagons provide a least-perimeter way to partition the plane into unit areas. Undergraduates recently obtained a partial extension to closed hyperbolic manifolds. The 3D Euclidean case remains open. The best tetrahedral tile was proved recently. (Despite what Aristotle said, the regular tetrahedron does not tile.)
We'll describe many such results and open questions.