At the 1962 ICM Shafarevich announced a conjecture regarding finiteness properties of families of smooth projective curves. It was confirmed in the geometric case by Parshin (1968) and Arakelov (1971), and in the arithmetic case by Faltings (1983). This conjecture is related to many other problems; perhaps the most famous one is the Mordell conjecture: by a very nice argument, now known as "Parshin's covering trick" the Mordell conjecture follows from the Shafarevich conjecture. . During the past 10 years years many results have been obtained with regard to higher dimensional generalizations of Shafarevich's conjecture in the geometric case. In this talk I will review the original conjecture, its possible generalizations, and the current knowledge about hyperbolicity, boundedness and rigidity of families of varieties of general type. In particular, I will discuss recent developments in each of those three directions (hyperbolicity, boundedness and rigidity). . One of these new developments is joint work with Stefan Kebekus and another one is joint work with Max Lieblich.