Superfluidity and superconductivity are remarkable
manifestations of quantum coherence at a macroscopic scale.
The dynamics of superfluids has dominated the study of these
systems for decades now, but a comprehensive theoretical
framework is still lacking. We introduce a local extension
of the time-dependent density functional theory to describe
the dynamics of fermionic superfluids. Within this approach
one can correctly represent vortex quantization, generation,
and dynamics, the transition from a superfluid to a normal
phase and  a number of other large amplitude collective
modes which are beyond the scope of two-fluid hydrodynamics,
Ginzburg-Landau and/or Gross-Pitaevskii  approaches. We
illustrate the power of this approach by studying the
generation of quantized vortices, vortex rings, vortex
reconnection, and transition from a superfluid to a normal
state in real time for a unitary Fermi gas. We predict the
emergence of a new qualitative phenomenon in superfluid
dynamics of gases, the existence of stable superfluidity
when the systems are stirred with velocities significantly
exceeding the nominal Landau critical velocity in these syste