We propose first a generalization of the Density Functional Theory. This theory leads to single-particle equations of motion with a quasilocal mean-field operator, which contains a quasiparticle position-dependent effective mass and a spin-orbit potential. The energy density functional is constructed using the extended Thomas-Fermi approximation. Some ground-state properties of doubly magic nuclei are considered within the framework of this approach. Calculations are performed using the finite-range Gogny DIS forces, and the results are compared with the exact Hartree-Fock calculations. Next, we present an extension of the density functional theory to include pairing correlations without formal violation of the particle-number condition. This problem, which is nonlocal, can be simplified by a suitable quasilocal reduction, which is also briefly discussed in this paper.