  Magnesium diboride MgB2 has been an extraordinarily hot research topic since
the discovery of superconductivity below 40 K, due to the vast amount of
unusual properties originating from the involvement in superconductivity of two
sets of bands that are of a different dimensionality. We review, with a strong
focus on results obtained by torque magnetometry, how this leads to a complex
behavior of the anisotropic superconducting state properties. The different
dimensionality of the two sets of bands is manifested in the upper critical
field Hc2 anisotropy, which was found to strongly decrease with increasing
temperature. While the angular dependence of Hc2 follows roughly the
predictions of anisotropic Ginzburg-Landau theory AGLT, small, but systematic
deviations were observed near Tc. This, and the temperature dependent
anisotropy, witness a breakdown of the AGLT description of MgB2, even close to
Tc. Theoretical calculations are in qualitative agreement with the observed
angle dependence, but suggest a much lower penetration depth anisotropy at low
temperatures. Reversible torque vs angle curves in the mixed state indicate
that the penetration depth anisotropy in intermediate fields has to be higher
than currently available theoretical estimates. The observed field dependent
effective anisotropies can be accounted for qualitatively by the faster
depression of superconductivity in the more isotropic bands by the applied
field. The vortex matter phase diagram is drawn, based on measurements of the
reversible and irreversible torque in the mixed state. In the irreversible
torque, a peak effect (PE) was observed in fields of about 0.85 Hc2. History
effects of the critical current density in the PE region suggest that the PE
signifies an order-disorder transition of vortex matter.
