On his way to the Mordell conjecture, Faltings proved that over any number field there are only finitely many smooth projective curves of given genus g > 1 with good reduction outside a given finite set of primes. This had been conjectured by Shafarevich at the ICM in 1962 along with a geometric finiteness statement for non-isotrivial families of complex curves. Similar finiteness results are expected to hold for a much larger class of higher-dimensional varieties, but very little is known. In the talk I will discuss recent work with Marco Maculan in which we prove the Shafarevich conjecture for a large class of canonically polarized varieties; the main geometric ingredient is a big monodromy theorem for families of smooth projective varieties with finite unramified Albanese morphism that extends our previous work with Javanpeykar and Lehn.