The current status of the theory of the electronic structure of polymers containing conjugated bonds is reviewed. Compounds having conjugated bonds have a number of remarkable physical properties, and they occupy a central point in quantum-mechanical studies. The energy of the first optical transition as a function of the molecular length shows unusual behavior. As the molecule is lengthened, the energy of the first transition approaches a finite value called the gap. Hence such a polymer is a semiconductor. A first attempt to explain the gap was based on the hypothesis of spontaneous alternation of bond lengths. The review presents this hypothesis and subjects it to a thorough critique. It has recently been shown that the mechanism that gives rise to the gap in these systems is interaction of electrons and an associated Mott metal-dielectric transition. This conclusion was first based on the unrestricted Hartree-Fock method, and later upon exact solutions for the one-dimensional problem. The article reviews these studies. Polymers with conjugated bonds also show non-trivial magnetic properties, in spite of their lack of d and f electrons. The magnetic properties of these compounds can be described on the basis of the one-dimensional Hamiltonian of Hubbard. Here an exact solution of the wave functions of the ground and excited states is found and the exact excitation spectrum is analyzed. The ground state of these polymers proves to be antiferromagnetic at absolute zero. The spin-wave spectrum begins at zero. Hence these systems show appreciable paramagnetism at finite temperatures. This review analyzes the relation of the paramagnetic susceptibility and the intensity of the EPR signal to the temperature and the polymer length. In conclusion, the review lists problems and questions in the theory of the electronic structure of conjugated polymers that await solution.