  Single crystals of MgB2 with a size up to 1.5x0.9x0.2 mm3 have been grown
with a high pressure cubic anvil technique. The crystal growth process is very
peculiar and involves an intermediate nitride, namely MgNB9. Single crystals of
BN and MgB2 grow simultaneously by a peritectic decomposition of MgNB9.
Magnetic measurements in fields of 1-5 Oe show sharp transitions to the
superconducting state at 37-38.6 K with width of ~0.5 K. The high quality of
the crystals allowed the accurate determination of magnetic, transport and
optical properties as well as scanning tunnelling spectroscopy (STS) and
decoration studies. Investigations of crystals with torque magnetometry show
that Hc2//c is very low (24 kOe at 15 K), while Hc2//ab increases up to 140 kOe
at 15 K. The upper critical field anisotropy gamma = Hc2//ab/ Hc2//c was found
to be temperature dependent (decreasing from 6 at 15 K to 2.8 at 35 K). The
effective anisotropy gamma_eff, as calculated from reversible torque data near
Tc, is field dependent (increasing roughly linearly from 2 in zero field to 3.7
in 10 kOe). The temperature and field dependence of the anisotropy can be
related to the double gap structure of MgB2 with a large two-dimensional gap
and small three-dimensional gap, the latter being rapidly suppressed in a
magnetic field. Torque magnetometry investigations show a pronounced peak
effect, indicating an order-disorder transition of vortex matter. Decoration
experiments and STS visualise a hexagonal vortex lattice. STS spectra evidence
two gaps (3 meV/6 meV) with direction dependent weight. Magneto-optic
investigations with H//c show a clear signature of the smaller of the two gaps,
disappearing in fields higher than Hc2//c.
