  A comprehensive theoretical study of direct laser excitation of a nuclear
state based on the density matrix formalism is presented. The nuclear clock
isomer $^{229\text{m}}$Th is discussed in detail, as it could allow for direct
laser excitation using existing technology and provides the motivation for this
work. The optical Bloch equations are derived for the simplest case of a pure
nuclear two-level system and for the more complex cases taking into account the
presence of magnetic sub-states, hyperfine-structure and Zeeman splitting in
external fields. Nuclear level splitting for free atoms and ions as well as for
nuclei in a solid-state environment is discussed individually. Based on the
obtained equations, nuclear population transfer in the low-saturation limit is
reviewed. Further, nuclear Rabi oscillations, power broadening and nuclear
two-photon excitation are considered. Finally, the theory is applied to the
special cases of $^{229\text{m}}$Th and $^{235\text{m}}$U, being the nuclear
excited states of lowest known excitation energies. The paper aims to be a
didactic review with many calculations given explicitly.
