  Let $\Omega\subset \mathbb R^2$ be a bounded planar domain, with piecewise
smooth boundary $\partial \Omega$. For $\sigma>0$, we consider the Robin
boundary value problem \[ -\Delta f =\lambda f, \qquad \frac{\partial
f}{\partial n} + \sigma f = 0 \mbox{ on } \partial \Omega \]
  where $ \frac{\partial f}{\partial n} $ is the derivative in the direction of
the outward pointing normal to $\partial \Omega$.
  Let $0<\lambda^\sigma_0\leq \lambda^\sigma_1\leq \dots $ be the corresponding
eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps
  \[ d_n(\sigma):=\lambda_n^\sigma-\lambda_n^0 . \] For a wide class of planar
domains we show that there is a limiting mean value, equal to $2{\rm
length}(\partial\Omega)/{\rm area}(\Omega)\cdot \sigma$ and in the smooth case,
give an upper bound of $d_n(\sigma)\leq C(\Omega ) n^{1/3}\sigma $ and a
uniform lower bound. For ergodic billiards we show that along a density-one
subsequence, the gaps converge to the mean value. We obtain further properties
for rectangles, where we have a uniform upper bound, and for disks, where we
improve the general upper bound.
