  (abbreviated) This article considers recent advances in the investigation of
the thermal and magnetic properties of integrable spin ladder models and their
applicability to the physics of real compounds. The ground state properties of
the integrable two-leg spin-1/2 and the mixed spin-(1/2,1) ladder models at
zero temperature are analyzed by means of the Thermodynamic Bethe Ansatz.
Solving the TBA equations yields exact results for the critical fields and
critical behaviour. The thermal and magnetic properties of the models are
investigated in terms of the recently introduced High Temperature Expansion
method, which is discussed in detail. It is shown that in the strong coupling
limit the integrable spin-1/2 ladder model exhibits three quantum phases: (i) a
gapped phase in the regime $H<H_{c1}$, (ii) a fully polarised phase for
$H>H_{c2}$, and (iii) a Luttinger liquid magnetic phase in the regime
$H_{c1}<H<H_{c2}$. The critical behaviour in the vicinity of the critical
points is of the Pokrovsky-Talapov type. The temperature-dependent thermal and
magnetic properties are directly evaluated from the exact free energy
expression and compared to known experimental results for a range of strong
coupling ladder compounds. Similar analysis of the mixed spin-(1/2,1) ladder
model reveals a rich phase diagram, with a 1/3 and a full saturation
magnetisation plateau within the strong antiferromagnetic rung coupling regime.
For weak rung coupling, the fractional magnetisation plateau is diminished and
a new quantum phase transition occurs. The phase diagram can be directly
deduced from the magnetisation curve obtained from the exact result derived
from the HTE. The thermodynamics of the spin-orbital model with different
single-ion anisotropies is also investigated.
