Advances in the development, fabrication and utilization of nano-scale materials, devices and systems are driven by the ability to control their structure and response on the nano-meter scale, that is at the level of atoms, molecules and clusters. This thesis deals with the theoretical treatment of the characteristic response to external perturbations of nanosystems, such as alkali-metal clusters, fullerenes and semiconductor quantum dots. It is shown that this response is determined by an interplay between electronic correlations and finite-size effects. Particular attention is devoted to the following many-body phenomena and collision processes: The polarization of the system in response to an approaching charged particle, such as electrons or protons. For a comparison with available experiments we considered the ionization channel. The role of exchange effects is revealed upon a comparison of ionization probability of the target by protons and electrons. We also investigate the single and the multiple photoionization of clusters. Both approximate many-body methods and exactly solvable problems are employed for the description of the electronic structure of the systems. We use the Hartree-Fock (HF) theory as a basis and account systematically for the electronelectron interactions in a perturbative way using the random phase approximation with exchange. The Hartree-Fock method, while formally well-established, represents severe technical difficulties in the application to the systems with a large number of electrons due to the presence of a huge number of the nonlocal exchange integrals. This essentially complicates the calculations, so that the finding a balance between the reliability of the results and the computational costs becomes an important, if not a decisive, issue. Therefore the right choice of the calculational procedure is the key question for solving nonlocal potential problems for systems with large number of electrons. To increase the efficiency of Hartree-Fock calculations, we develop in this thesis a new conceptual framework, based on the extension of the variable phase approach (VPA) to nonlocal potentials. In the VPA the solution of quantum-mechanical problems is formulated in terms of observables, e.g. scattering phase shifts and scattering amplitudes, rather than by means of the wave functions. We also propose the efficient finite-difference scheme for a numerical implementation. The advantage of this scheme stems from the fact that for the calculation of the Hartree-Fock integrals only the diagonal part of the nonlocal one-electron potential is required. Based on the developed VPA scheme, we reformulate the Hartree-Fock problem for nano-size systems. The ionization of many-electron systems by a charged particle is a fundamental physical process, allowing a direct information on the correlated electron dynamics. The fluctuations of the electronic charge density of the clusters, accounted for within the random phase approximation with exchange, are shown to suppress the single ionization channel in the low-energy region, while at large impact energies they are not essential. This tendency is in full agreement with the experimental data for fullerene clusters. In contrast, the calculations performed without the account of the dynamic screening were unable to reproduce the experimental data, giving a wrong low-energy behavior of the ionization cross section. Our first principle numerical results are interpreted with the help of the Thomas-Fermi model of screening. The qualitative conclusion of this study is that the change of the effective radius of the inter-electron interaction does not lead to a simple scaling of the ionization probability, as it could be supposed, but modifies also its energy dependence. In addition, we investigate the interplay between quantum size and nonlocal screening effects by tracing the changes in the ionization cross sections for Li clusters with an increasing cluster radius. The particle-hole excitations proved to play a decisive role in the evolution of the ionization cross sections of the clusters with the increase of their size and, consequently, with the increase of the number of delocalized electrons. Actually, we observe that the inclusion of the dynamic screening leads to the flattening of the ionization cross section as the cluster size grows, while the disregard of this phenomenon gives the opposite tendency. What is less obvious, the energy position of the maximum in the cross section also shifts in the opposite directions. It is hoped that these striking size effects will stimulate further experimental measurements. A specific sort of correlation effects arising from the exchange interaction is traced down via a comparison of the proton and the electron impact ionization of C60. The absence of the exchange between the proton and the target electrons qualitatively plays the same role as the disregard of the particle-hole (de)excitations in the many-electron system. In particular, the ~ 30% increase of the proton impact ionization probability as compared to the electron impact is observed. For the description of the multiple ionization of fullerenes by a single photon, the statistical energy deposition model is used for the first time. Usually, the calculations based on this model employed adjustable or semi-empirical parameters for the estimation of the single ionization transition amplitude, which enters the expression for the multiple ionization probability. We improve this step by performing the first principle calculations of this quantity. In doing this we get a natural access to the material, structural and energy dependence of the multiple ionization processes. For the first time the probabilities of different multiple photoionization reactions in C60 are estimated and compared. The diffraction of the photoelectron wave on the fullerene shell is shown to be a prominent feature of these processes. One of the few exactly solvable quantum-mechanical problems is the problem of two electrons, coupled via bare Coulomb repulsion, in the parabolic confinement. Using the exact two-electron wave function, the double photoionization cross section of a quantum dot is derived and the various angular and energy spectra are calculated. The mapping of the two-electron density onto the double photoionization spectra is suggested as a promising tool for the visualization of subtle correlation effects. The results of this work are applicable to problems dealing with arbitrary nonlocal or momentum-dependent potentials, to arbitrary systems of integro-differential equations and as a theoretical background for the experimental investigations of the influence of finite size effects on the strength of the electronic correlation. Most of our results can be directly verified experimentally using modern many-particle coincidence spectroscopic techniques.