Our nearest star the Sun exhibits prominent 11-year cycles of magnetic activity, with emerging sunspots following reasonably well defined rules as the cycles proceed. The origin of these strong magnetic fields must rest with dynamo processes resulting from the interaction of convection, rotation, shear and magnetism within the highly turbulent convection zone occupying the outer 30% by radius of this star just below its surface. Helioseismology has allowed us to probe some of the subsurface flows and the differential rotation of that zone, providing important guidance and constraints for theoretical dynamo models. We will discuss results achieved with our 3-D Anelastic Spherical Harmonic (ASH) global simulations that are revealing how strong toroidal wreaths of magnetism can be built within the convection zone itself, and how portions can become unstable and rise toward the surface, likely becoming emerging flux structures. Many of the models possess cycles in which the global fields can reverse the polarity of their wreaths. Our most recent models can have strikingly regular magnetic cycles that are interrupted by a period of relative quiescence, not unlike observed grand minina, before resuming their cycling. There are major challenges in studying these turbulent processes, yet the 3-D modeling is helping to unravel dynamical elements that must be at work deep within our nearest star.