  We provide general upper and lower bounds for the Gromov-Hausdorff distance
$d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)$ between spheres $\mathbb{S}^m$ and
$\mathbb{S}^n$ (endowed with the round metric) for $0\leq m< n\leq \infty$.
Some of these lower bounds are based on certain topological ideas related to
the Borsuk-Ulam theorem. Via explicit constructions of (optimal)
correspondences we prove that our lower bounds are tight in the cases of
$d_{\mathrm{GH}}(\mathbb{S}^0,\mathbb{S}^n)$,
$d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^\infty)$,
$d_{\mathrm{GH}}(\mathbb{S}^1,\mathbb{S}^2)$,
$d_{\mathrm{GH}}(\mathbb{S}^1,\mathbb{S}^3)$ and
$d_{\mathrm{GH}}(\mathbb{S}^2,\mathbb{S}^3)$. We also formulate a number of
open questions.
