  The experimentally established phase diagram of the half-filled Hubbard model
features the existence of three distinct finite-temperature regimes, separated
by extended crossover regions. A number of crossover lines can be defined to
span those regions, which we explore in quantitative detail within the
framework of dynamical mean-field theory. Most significantly, the high
temperature crossover between the bad metal and Mott-insulator regimes displays
a number of phenomena marking the gradual development of the Mott insulating
state. We discuss the quantum critical scaling behavior found in this regime,
and propose methods to facilitate its possible experimental observation. We
also introduce the concept of {\em quantum Widom lines} and present a detailed
discussion that highlights its physical meaning when used in the context of
quantum phase transitions.
