  The quantum measurement problem, understanding why a unique outcome is
obtained in each individual experiment, is tackled by solving models. After an
introduction we review the many dynamical models proposed over the years. A
flexible and rather realistic model is introduced, describing the measurement
of the $z$-component of a spin through interaction with a magnetic memory
simulated by a Curie--Weiss magnet, including $N \gg1$ spins weakly coupled to
a phonon bath. Initially prepared in a metastable paramagnetic state, it may
transit to its up or down ferromagnetic state, triggered by its coupling with
the tested spin, so that its magnetization acts as a pointer.
  A detailed solution of the dynamical equations is worked out. Conditions are
found, which ensure that the process satisfies the features of ideal
measurements. Various imperfections are discussed, as well as attempts of
incompatible measurements. The first steps consist in the solution of the
Hamiltonian dynamics for the spin-apparatus density matrix $D(t)$. On a longer
time scale, the trend towards equilibrium of the magnet produces a final state
$D(t_{\rm f})$ that involves correlations between the system and the
indications of the pointer, thus ensuring registration. A difficulty lies in a
quantum ambiguity: There exist many incompatible decompositions of the density
matrix $\scriptD(t_{\rm f})$. This difficulty is overcome by dynamics due to
suitable interactions within the apparatus.
  Any subset of runs thus reaches over a brief delay a stable state which
satisfies the same hierarchic property as in classical probability theory.
Standard quantum statistical mechanics alone appears sufficient to explain the
occurrence of a unique answer in each run. Finally, pedagogical exercises are
proposed while the statistical interpretation is promoted for teaching.
[Abridged]
