Consistency with quantum gravity can impose non-trivial constraints at low energies, even if the Planck scale is at very high energy. The Swampland program aims to determine the constraints that an effective field theory must satisfy to be consistent with a UV embedding in a quantum gravity theory. One of the most important swampland conditions is the presence of infinite towers of states becoming massless at the weak coupling/large field limits. This has been extensively tested in string theory compactifications, but a bottom-up explanation was missing. In this talk I will provide a possible explanation based on finiteness of black hole entropy. I will also explain how several wampland criteria, including the Weak Gravity Conjecture, Distance Conjecture and bounds on the finiteness of the quantum gravity vacua, may be more fundamentally a consequence of the finiteness of quantum gravity amplitudes.