  In this work, we study the issue of the past and future cosmological
singularities in the context of the fourth-order conformal Weyl gravity. In
particular, we investigate the emergent universe scenario proposed by Ellis
{\it et al}, and find the stability conditions of the corresponding Einstein
static state using the fixed point approach. We show that depending on the
values of the parameters of the conformal Weyl gravity theory, there are
possibilities for having initially stable emergent states for an FRW universe
with both the positive and negative spatial curvatures. This represents that
the conformal Weyl gravity can be free of the initial singularity problem.
Then, following Barrow {\it et al}, we address the possible types of the
finite-time future cosmological singularities. We discuss how these
singularities also can be avoided in the context of this theory.
