We have investigated processes of ionization, energy absorption and subsequent explosion of atomic and molecular clusters under intense laser illumination using numerical as well as analytical methods. In particular, we focused on the response of composite clusters, those consisting of different atomic elements, to intense light pulses. Another major theme is the effect of the molecular structure of clusters on their Coulomb explosion. The action of intense laser pulses on clusters leads to fundamental, irreversible changes: they turn almost instantaneously into nanoplasmas and subsequently disintegrate into separate ions and electrons. Due to this radical transformation, remarkable new features arise. Transient cluster nanoplasmas are capable of absorbing enormous amounts of laser energy. In some cases more than 90 % of incident laser energy is absorbed by a gas of clusters with a density much smaller than that of a solid. After the efficient absorption, the energy is transformed into production of energetic ions, electrons, photons, and even neutrons. Composite clusters show especially interesting behavior when they interact with intense laser pulses. Nanoplasmas formed in composite clusters may absorb even more laser energy, than those formed in homogeneous clusters, as we demonstrate in this work. One of the most important results of this thesis is the identification of a novel type of plasma resonance. This resonance is enabled by an unusual ellipsoidal shape of the nanoplasma created during the ionization process in a helium droplet doped with just a few xenon atoms. In contrast to the conventional plasma resonance, which requires significant ion motion, here, the resonant energy absorption occurs at a remarkably fast rate, within a few laser cycles. Therefore, this resonance is not only the most efficient (like the conventional resonance), but also, perhaps, the fastest way to transfer laser energy to clusters. Recently, dedicated experimental studies of this effect were performed at the Max Planck Institute in Heidelberg. Their preliminary results confirm our prediction of a strong, avalanche-like ionization of the helium droplet with a small xenon cluster inside. A conventional plasma resonance, which relies on the cluster explosion, also exhibits interesting new properties when it occurs in a composite xenon-helium cluster with a core-shell geometry. We have revealed an intriguing double plasma resonance in this system. This was the first theoretical study of the influence of the helium embedding on the laser- driven nanoplasma dynamics. Our results demonstrate the important role of the interaction between xenon and helium parts of the cluster. Understanding this interaction is necessary in order to correctly interpret the experimental results. We have elucidated several important properties of Coulomb explosion in atomic and molecular clusters. Specifically, it was found that the kinetic energy distribution of ions after the Coulomb explosion of an atomic cluster is quite similar to the initial potential energy distribution of ions and is only weakly influenced by ion overtake effects, as was believed before. For the case of molecular hydrogen clusters, we have shown that the alignment of molecules inside the cluster affects its Coulomb explosion. Investigation of the dynamical processes in composite and molecular clusters induced by intense laser pulses is a step towards understanding them in more complex nano-objects, such as biomolecules or viruses. This is of great interest in the context of x-ray diffractive imaging of biomolecules with atomic resolution, which is one of the main goals of new x-ray free electron laser facilities.