Beginning in the late 1980s, many unexpected superconductors have emerged. They raised not only the transition temperature limit but, just as importantly, the theoretical challenges beyond the BCS theory. Following the previous article on historical perspective of low-temperature specific heat of superconductors prior to 1986, we review briefly here some calorimetric studies on representative high-Tc and other novel superconductors discovered during the new era. Major emphasis is centered on special features of each set of temperature- and magnetic field-dependent specific heat data being fitted to select theoretical models. The models may involve multi-gap or a certain type of order-parameter symmetry.