  Hydrogen-like light muonic ions, in which one negative muon replaces all the
electrons, are extremely sensitive probes of nuclear structure, because the
large muon mass increases tremendously the wave function overlap with the
nucleus. Using pulsed laser spectroscopy we have measured three 2S-2P
transitions in the muonic helium-3 ion ($\mu^3$He$^+$), an ion formed by a
negative muon and bare helium-3 nucleus. This allowed us to extract the Lamb
shift $E(2P_{1/2}-2S_{1/2})= 1258.598(48)^{\rm exp}(3)^{\rm theo}$ meV, the 2P
fine structure splitting $E_{\rm FS}^{\rm exp} = 144.958(114)$ meV, and the
2S-hyperfine splitting (HFS) $E_{\rm HFS}^{\rm exp} = -166.495(104)^{\rm
exp}(3)^{\rm theo}$ meV in $\mu^3$He$^+$. Comparing these measurements to
theory we determine the rms charge radius of the helion ($^3$He nucleus) to be
$r_h$ = 1.97007(94) fm. This radius represents a benchmark for few nucleon
theories and opens the way for precision tests in $^3$He atoms and $^3$He-ions.
This radius is in good agreement with the value from elastic electron
scattering, but a factor 15 more accurate. Combining our Lamb shift measurement
with our earlier one in $\mu^4$He$^+$ we obtain $r_h^2-r_\alpha^2 =
1.0636(6)^{\rm exp}(30)^{\rm theo}$ fm$^2$ to be compared to results from the
isotope shift measurements in regular He atoms, which are however affected by
long-standing tensions. By comparing $E_{\rm HFS}^{\rm exp}$ with theory we
also obtain the two-photon-exchange contribution (including higher orders)
which is another important benchmark for ab-initio few-nucleon theories aiming
at understanding the magnetic and current structure of light nuclei.
