The Lemaitre–Tolman class of cosmological models (spherically symmetric inhomogeneous metrics obeying the Einstein equations with a dust source) is derived and discussed in much detail, from the point of view of its geometry and its applications to cosmology. It is shown that these metrics can be used to describe the formation of cosmic voids and of galaxy clusters out of small perturbations of homogeneity at the time of emission of the cosmic microwave background radiation. Apparent horizons for central and noncentral observers, the formation of black holes, the existence and avoidance of shell crossings, the equations of redshift and the generation and meaning of blueshift are discussed. A simple example of a shell focussing singularity is derived. Among the cosmological applications are: solving the horizon problem without inflation, mimicking the accelerating expansion of the Universe by mass-density inhomogeneities in a decelerating model, drift of light rays, lagging cores of Big Bang, misleading conclusions drawn from observed mass distribution in redshift space.