Magnetohydrodynamic (MHD) mechanisms often are invoked as a model for the launching (initial acceleration) and collimation of astrophysical jets. There has been a growing recognition in recent years, however, that the influence of strong magnetic field within the jet may extend well beyond the central engine into the region where the jet freely propagates, influenced only by internal and ambient hydrodynamic (HD) and MHD forces. This is particularly evident in observations of jets in AGNs, QSOs, winds from pulsars, and GRBs. Strongly magnetized jets, particularly those with a strong toroidal component encircling the collimated flow, are often referred to as "current-carrying" or "Poynting flux-dominated" (PFD) jets. A large electric current flowing parallel to the flow is responsible for generating a strong, tightly-wound helical magnetic field. Continued rotation of the magnetized plasma in a black hole - accretion disk system drives a "barber pole"-like, torsional Alfven wave (TAW) forward in the direction of the jet flow, carrying electromagnetic (EM) energy and further accelerating the plasma. In a PFD jet, the EM energy carried by this TAW can greatly exceed the kinetic/matter energy fluxes in the HD part of the flow. Origin of the underlying large-scale magnetic field in the AGN jets is still our mystery and one of the biggest open issues. However, it is also true that the bulk acceleration up to a relativistic speed with huge Lorentz factors (10-1000) and the high collimation can successively occur by the asymptotic energy conversion from Poynting to kinetic energy flux in the relativistic MHD flows along the large-scale magnetic field. Several aspects such as the acceleration, collimation, and stability properties of the PFD jet will be discussed in terms of both theoretical and numerical approaches. We finally will exhibit our recent numerical results of the current-driven instability of PFD jets to model the "wiggles/kinks" seen in the VLBI scales AGN jets.