  A locally conformally K\"ahler (lcK) manifold is a complex manifold $(M,J)$
together with a Hermitian metric $g$ which is conformal to a K\"ahler metric in
the neighbourhood of each point. In this paper we obtain three classification
results in locally conformally K\"ahler geometry. The first one is the
classification of conformal classes on compact manifolds containing two
non-homothetic K\"ahler metrics. The second one is the classification of
compact Einstein locally conformally K\"ahler manifolds. The third result is
the classification of the possible (restricted) Riemannian holonomy groups of
compact locally conformally K\"ahler manifolds. We show that every locally (but
not globally) conformally K\"ahler compact manifold of dimension $2n$ has
holonomy $\mathrm{SO}(2n)$, unless it is Vaisman, in which case it has
restricted holonomy $\mathrm{SO}(2n-1)$. We also show that the restricted
holonomy of a proper globally conformally K\"ahler compact manifold of
dimension $2n$ is either $\mathrm{SO}(2n)$, or $\mathrm{SO}(2n-1)$, or
$\mathrm{U}(n)$, and we give the complete description of the possible solutions
in the last two cases.

Differential Geometry