We present a weakly nonlocal gravitational theory unitary and finite at quantum level in the quantum field theory framework. As a consequence of finiteness, there is no Weyl anomaly and the theory turns out to be conformal invariant at classical as well at quantum level. Therefore, nonlocal quantum gravity is a conformal invariant theory in the spontaneously broken phase of the Weyl symmetry. As an application, Weyl conformal symmetry solves the black hole's singularity issue and cosmological singularity problem, otherwise unavoidable in a generally covariant local or non-local gravitational theory. Following and extending the seminal paper by Narlikar and Kembhavi, we provide explicit examples of singularity-free black hole exact solutions. The absence of divergences is based on the finiteness of the curvature invariants and on showing that the new types of black holes are geodesically complete. Indeed, no massive or massless particle can reach the former singularity in a finite amount of proper time or of affine parameter.