The strict hyperbolization process of R. Charney and M. Davis produces a large and rich class of negatively curved spaces (in the geodesic sense). This process is based on an earlier version introduced by M. Gromov and later studied by M. Davis and T. Januszkiewicz. If M is a manifold its Charney-Davis strict hyperbolization is also a manifold, but the negatively curved metric obtained is far from being Riemannian because it has a large and complicated set of singularities. We will discuss whether these singularities can be removed, so that this process can be done smoothly.