Semiconductor nanostructures are applied in various electronic and optoelectronic devices. As miniaturization of these devices progresses, a microscopic treatment of correlations between excited carriers is essential for understanding and describing the governing physics. We investigate two different types of semiconductor nanostructures, which have each received considerable attention over the last years. These are self-assembled InGaAs quantum dots (QDs) on the one hand and atomic monolayers of MoS2 on the other hand. Self-assembled semiconductor QDs are used as active material in conventional lasers and as efficient non-classical light sources with applications in quantum information. As they can confine a small number of carriers in localized stats with discrete energies, it is questionable to neglect correlations between the carriers when describing their dynamics. We analyze the influence of carrier correlations in a single QD on Coulomb scattering processes, which are due to the contact with a quasi-continuum of wetting-layer (WL) states. Results obtained from a Boltzmann equation are compared with the fully correlated dynamics governed by a von-Neumann-Lindblad equation. In a first step, we take into account correlations generated by the exact treatment of Pauli blocking due to the contributing QD carrier configurations. Subsequently, we include correlations generated by energy renormalizations due to Coulomb interaction between the QD carriers. It is shown that at low WL carrier densities, neither Pauli correlations nor Coulomb correlations can be safely neglected, if the dynamics of single-particle states in the QD are to be predicted qualitatively and quantitatively. In the high-density regime, both types of correlations play a lesser role and thus a description of carrier dynamics by a Boltzmann equation becomes reliable. Furthermore, the efficiency of WL-assisted scattering processes as well as scattering-induced dephasing rates depending on the WL carrier density are discussed. Subsequently, experimental results for the carrier capture and relaxation dynamics in QDs are analyzed using a microscopic theory including also carrier-LO-phonon interaction. Time-resolved differential transmission changes of the QD transitions after ultrafast optical excitation of the barrier states are studied in a wide range of carrier temperatures and excitation densities. The measurements can be explained by QD polaron scattering and their excitation-dependent renormalization due to additional Coulomb scattering processes. Results of a configuration-picture and single-particle picture description, both with non-perturbative transition rates, show good agreement with the experiments while Boltzmann scattering rates lead to a different excitation density and temperature dependence. Monolayer MoS2 is the most prominent member of the class of two-dimensional transition-metal dichalcogenides and exhibits a direct band gap, making it a promising candidate for optical applications. We study the ground-state and finite-density optical response of MoS2 by solving the semiconductor Bloch equations, using ab-initio band structures and Coulomb interaction matrix elements. Spectra for excited carrier densities up to 10^13/cm^2 reveal a redshift of the excitonic ground-state absorption, whereas higher excitonic lines are found to disappear successively due to Coulomb-induced band-gap shrinkage of more than 500 meV and binding-energy reduction. Strain-induced band variations lead to a redshift of the lowest exciton line by approx. 110 meV/% and change the direct transition to indirect while maintaining the magnitude of the optical response
