A brief introduction to the notorious "cosmological constant problem" is given. Then, a particular approach is discussed, which has been developed by Volovik and the present speaker over the last years and which goes under the name of q-theory. This approach provides a possible solution of the main cosmological constant problem, why is |Lambda|^(1/4) negligible compared to the energy scales of the electroweak standard model (not to mention the Planck energy)? The next problem is, of course, the small but nonzero value of the actual cosmological constant, responsible for the observed "accelerating universe." This problem can also be addressed in the framework of q-theory. In fact, the observed value Lambda ~ (meV)^4 may correspond to the remnant vacuum energy density of dynamical processes taking place at a cosmic age set by the mass scale M ~ E_ew of ultramassive particles with electroweak interactions. A first estimate of the required value of the energy scale E_ew ranges from 3 to 9 TeV. If correct, this estimate implies the existence of new TeV-scale physics beyond the standard model.