  We consider the structure and physical properties of specific classes of
neutron, quark, and Bose-Einstein Condensate stars in the conformally invariant
Weyl geometric gravity theory. The basic theory is derived from the simplest
conformally invariant action, constructed, in Weyl geometry, from the square of
the Weyl scalar, the strength of the Weyl vector, and a matter term,
respectively. The action is linearized in the Weyl scalar by introducing an
auxiliary scalar field. To keep the theory conformally invariant the trace
condition is imposed on the matter energy-momentum tensor. The field equations
are derived by varying the action with respect to the metric tensor, Weyl
vector field and scalar field. By adopting a static spherically symmetric
interior geometry, we obtain the field equations, describing the structure and
properties of stellar objects in Weyl geometric gravity. The solutions of the
field equations are obtained numerically, for different equations of state of
the neutron and quark matter. More specifically, constant density stellar
models, and models described by the stiff fluid, radiation fluid, quark bag
model, and Bose-Einstein Condensate equations of state are explicitly
constructed numerically in both general relativity and Weyl geometric gravity,
thus allowing an in depth comparison between the predictions of these two
gravitational theories. As a general result it turns out that for all the
considered equations of state, Weyl geometric gravity stars are more massive
than their general relativistic counterparts. As a possible astrophysical
application of the obtained results we suggest that the recently observed
neutron stars, with masses in the range of 2$M_{\odot}$ and 3$M_{\odot}$,
respectively, could be in fact conformally invariant Weyl geometric neutron or
quark stars.
