We build a case for the persistent strong coronal heating in active regions and the pervasive quasi-steady heating of the corona in quiet regions and coronal holes being driven in basically the same way as the intense transient heating in solar flares: by explosions of sheared magnetic fields in the cores of initially closed bipoles. We begin by summarizing the observational case for exploding sheared core fields being the drivers of a wide variety of flare events, with and without coronal mass ejections. We conclude that the arrangement of an event's flare heating, whether there is a coronal mass ejection, and the time and place of the ejection relative to the flare heating are all largely determined by four elements of the form and action of the magnetic field: (1) the arrangement of the impacted, interacting bipoles participating in the event, (2) which of these bipoles are active (have sheared core fields that explode) and which are passive (are heated by injection from impacted active bipoles), (3) which core field explodes first, and (4) which core-field explosions are confined within the closed field of their bipoles and which ejectively open their bipoles. We then apply this magnetic-configuration framework for flare heating to the strong coronal heating observed by the Yohkoh Soft X-ray Telescope in an active region with strongly sheared core fields observed by the MSFC vector magnetograph. All of the strong coronal heating is in continually microflaring sheared core fields or in extended loops rooted against the active core fields. Thus, the strong heating occurs in field configurations consistent with the heating being driven by frequent core-field explosions that are smaller but similar to those in confined flares and flaring arches. From analysis of the thermal and magnetic energetics of two selected core-field microflares and a bright extended loop, we find that (1) it is energetically feasible for the sheared core fields to drive all of the coronal heating in the active region via a staccato of magnetic microexplosions, (2) the microflares at the feet of the extended loop behave as the flares at the feet of flaring arches in that more coronal heating is driven within the active bipole than in the extended loop, (3) the filling factor of the X-ray plasma in the core field microflares and in the extended loop is approximately 0.1, and (4) to release enough magnetic energy for a typical microflare (10^27 - 10^28 erg), a microflaring strand of sheared core field need expand and/or untwist by only a few percent at most. Finally, we point out that (1) the field configurations for strong coronal heating in our example active region (i.e., neutral-line core fields, many embedded in the feet of extended loops) are present in abundance in the magnetic network in quiet regions and coronal holes, and (2) it is known that many network bipoles do microflare and that many produce detectable coronal heating. We therefore propose that exploding sheared core fields are the drivers of most of the heating and dynamics of the solar atmosphere, ranging from the largest and most powerful coronal mass ejections and flares, to the vigorous microflaring and coronal heating in active regions, to the multitude of fine-scale explosive events in the magnetic network. The low-lysing exploding core fields in the network drive microflares, spicules, global coronal heating, and ,consequently, the solar wind.