In the 1970s and 1980s, W. Thurston's seminal work on mapping class groups and hyperbolization of surface bundles over the circle inspired numerous research directions. In particular, in the 1990s the following problem started gaining consideration attention: Can one construct hyperbolic surface bundles over surfaces? Such surface bundles are atoroidal, and this led to a follow up: Can one construct atoroidal surface bundles over surfaces? In this talk I'll recall Thurston's work, provide a general discussion of surface bundles and mapping class groups, describe the evolution of the questions above, and discuss my work with Autumn Kent producing infinitely many atoroidal surface bundles, "many" of which fail to be hyperbolic manifolds.