While the process that converts implosion into explosion in core collapse supernovae is poorly understood, their observed asphericity provides new constraints on the physics of these events. Since pulsars are rotating and magnetized neutron stars, there is no doubt that rotation and magnetic fields are inherent to the exploding engine. We have shown that magnetic field amplification is an inevitable by-product of the differential rotation that accompanies core-collapse. We performed 1D core-collapse simulations of rotating iron cores with various rotational profiles and velocities. We found that differential rotation was a generic feature of rotating iron core collapse. As a result, the magnetorotational instability (MRI) generates magnetic fields of order 10¹⁵⁻¹⁷ G in a few tens of milliseconds where the negative shear is the strongest. Although magnetic fields of order 10¹⁵⁻¹⁷ G are very strong, they are not strong enough to modify the equation of state of degenerate electron gas near the proto-neutron star. The corresponding MHD luminosity available is ~10⁵² erg s⁻¹, which can modify the explosion dynamics if the power is sustained for a fraction of a second. When rotational effects are included, we found that there is a critical iron core rotation rate that gives the most rapidly rotating proto-neutron star, faster than which the rotational velocity of the proto-neutron star decreases due to centrifugal support. This non-monotonic behavior of post-collapse core rotation suggests that the progenitor of the most rapidly rotating proto-neutron star is not the most rapidly rotating iron core, but that those iron cores with nearly the critical initial rotation rate may produce the maximum proto-neutron star rotation, the strongest magnetic fields, and the most robust supernova explosions. Even small rotation may induce non-axisymmetric instabilities, which drive magneto-acoustic flux in to the mantle, transporting enegy out of the proto-neutron star to the region near the stalled shock. Further implications for rotation and magnetic fields, pulsars and magnetars, and jet formation mechanisms are discussed. / text