Here, the most relevant aspect is the treatment of the NAKE and the associated NAKP, and novel developments of approximations for these are presented in Section 4.1. The treatment of the non-additive exchange-correlation energy (NAXC) has also attracted ) for the case of two subsystems (A and B), reformulated to highlight the subsystem framework (i.e., the total energy is given as a sum of Kohn-Sham energy expressions for the two subsystem densities plus electrostatic, exchange-correlation, and non-additive kinetic terms mediating the interaction). some interest, and strategies for approximating the NAXC energy in the case of non-local xc-functionals and for improving upon conventional xc approximations within sDFT are discussed in Section 4.2. The use of potential reconstruction techniques in order to avoid approximations for the NAKE altogether are presented in Section 4.3. Surveys of further fundamental aspects related to sDFT/FDE can be found in our original article and in the review by Wesolowski and coworkers . An instructive comparative discussion of the fundamentals of sDFT/FDE in relation to other density-based subsystem and embedding approaches is provided by Pavanello and coworkers (see Table in Ref. ). For recent development in partition DFT, which shares several features with sDFT but allows for fractional numbers of electrons per subsystem and enforces a common embedding potential for all subsystems, we refer to the work of Wasserman and coworkers . Finally, the use of sDFT/FDE for accessing specific spin states and broken-symmetry sDFT approaches are presented in Section 4.4. A related review of developments and applications of sDFT/FDE for spin densities and spin-related properties is given in Ref. Second, applications of sDFT/FDE cover a wide range of different types of chemical systems, and specific variants and extensions have been developed that are tailored to certain types of applications. These will be covered in Section 5. Most importantly, there is a continuum between sDFT, in which all subsystems are treated at the same footing, and FDE, in which one focusses on one specific subsystem of interest. In the latter case, numerous strategies are available for choosing the frozen electron densities. Moreover, sDFT/FDE can be further combined with other embedding approaches in sophisticated multilevel schemes.