Aplanatic optics were invented over a century ago, motivated principally to achieve high-fidelity imaging in telescopes, microscopes and cameras. Aplanats are designed to completely eliminate the two leading orders of geometric aberration - spherical and comatic - and the simplest designs comprise two contours that can be reflective and/or refractive. Aplanats of high radiative efficiency can also approach the thermodynamic limit to flux concentration and light collimation - of particular value in nonimaging applications such as solar energy collection, light-emitting-diode collimation, and infrared technology. Recently, it was discovered that the original aplanatic mirrors and lenses cover only a small spot in a rich landscape of fundamental categories of optical devices, which opened a broad spectrum of powerful new designs. In this presentation we review these advances, and summarize the complete classification schemes that have now been elucidated for aplanats. They include examples of practical designs for achieving radiative transfer near the thermodynamic limit in flux concentration and irradiation applications, based on dual-mirror, dual-contour lens and lens-mirror combinations. The representative designs that are illustrated also include the most recent progress in Fresnel (faceted) aplanats, motivated by the quest for progressively more compact optical systems, as well as examples of hybrid designs – combining aplanats of different classifications for enhanced performance.