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The first two eigenvalues of the Robin Laplacian are investigated along withtheir gap and ratio. Conjectures by various authors for arbitrary domains aresupported here by new results for rectangular boxes.  Results for rectangular domains include that: the square minimizes the firsteigenvalue among rectangles under area normalization, when the Robin parameter$\alpha \in \mathbb{R}$ is scaled by perimeter; that the square maximizes thesecond eigenvalue for a sharp range of $\alpha$-values; that the line segmentminimizes the Robin spectral gap under diameter normalization for each $\alpha\in \mathbb{R}$; and the square maximizes the spectral ratio among rectangleswhen $\alpha>0$. Further, the spectral gap of each rectangle is shown to be anincreasing function of the Robin parameter, and the second eigenvalue isconcave with respect to $\alpha$.  Lastly, the shape of a Robin rectangle can be heard from just its first twofrequencies, except in the Neumann case.

The Robin Laplacian - spectral conjectures, rectangular theorems