  The spherically symmetric thin shells are the nearest generalizations of the
point-like particles. Moreover, they serve as the simple sources of the
gravitational fields both in General Relativity and much more complex quadratic
gravity theories. We are interested in the special and physically important
case when all the quadratic in curvature tensor (Riemann tensor) and its
contractions (Ricci tensor and scalar curvature) terms are present in the form
of the square of Weyl tensor. By definition, the energy-momentum tensor of the
thin shell is proportional to Dirac delta-function. We constructed the theory
of the spherically symmetric thin shells for three types of gravitational
theories with the shell: (1) General Relativity; (2) Pure conformal (Weyl)
gravity where the gravitational part of the total Lagrangian is just the square
of the Weyl tensor; (3) Weyl+Einstein gravity. The results are compared with
these in General Relativity (Israel equations). We considered in details the
shells immersed in the vacuum. Some peculiar properties of such shells are
found. In particular, for the traceless (= massless) shells it is shown that
their dynamics can not be derived from the matching conditions and, thus, is
completely arbitrary. On the contrary, in the case of the Weyl+Einstein gravity
the trajectory of the same type of shell is completely restored even without
knowledge of the outside solution.
