The Fermi surface is an abstract object in the reciprocal space for a material lattice, enclosing the set of all electronic band states that are filled according to the Pauli principle. Its topology is dictated by the underlying material structure and its volume is the carrier density in the material. The Fermi surface is central to predictions of thermal, electrical, magnetic, optical and superconducting properties in metallic systems. Density functional theory is a first-principles method used to calculate the occupied-band energies and the Fermi energy. In this talk, we survey several key facts about Fermi surfaces in complex systems, where a proper theoretical understanding is still lacking. We address some critical difficulties as to whether the Fermi surface is a ground state property and concerning its stability in strongly correlated systems.