  We review complicated problems in the Lifshitz theory describing the Casimir
force between real material plates made of metals and dielectrics including
different approaches to their resolution. It has been shown that both for
metallic plates with perfect crystal lattices and for any dielectric plates the
Casimir entropy calculated in the framework of the Lifshitz theory violates the
Nernst heat theorem when the well approved dielectric functions are used in
computations. The respective theoretical Casimir forces are excluded by the
measurement data of numerous precision experiments. In the literature this
situation received the names of the Casimir puzzle and the Casimir conundrum
for the cases of metallic and dielectric plates, respectively. The review
presents a summary of the main facts on this subject on both theoretical and
experimental sides. Next, we discuss the main approaches proposed in the
literature in order to bring the Lifshitz theory in agreement with the
measurement data and with the laws of thermodynamics. Special attention is paid
to the recently suggested spatially nonlocal Drude-like response functions
which take into account the relaxation properties of conduction electrons, as
does the standard Drude model, but lead to the theoretical results in agreement
with both thermodynamics and the measurement data through the alternative
response to quantum fluctuations off the mass shell. Further advances and
trends in this field of research are discussed.
