High-precision atomic calculations combined with experiments of matching accuracy provide an excellent opportunity to test our understanding of atomic structure and properties as well as the many-body atomic theories. The relativistic all-order method, which is a linearized version of the coupled-cluster singles-doubles method, has proven to yield high precision results for a variety of atomic properties. In this thesis, we study the atomic properties of neutral atoms and ions by means of the relativistic all-order method. The lifetimes and ground state static polarizabilities of a singly ionized barium atom are studied in comparison with the isoelectronic neutral cesium atom and with a singly ionized calcium atom. The lifetimes of a number of excited states in atomic potassium, rubidium, and francium are theoretically calculated and compared with the available experimental data. The magnetic dipole hyperfine constant of the 9S_1/2 state in ²¹⁰Fr is calculated and the result is combined with the experimental one to extract the value of the 210Fr nuclear magnetic moment. Another part of the thesis work focuses on the development and implementation of an extension of the currently used all-order singles-doubles (SD) method to include all valence triple excitations in an iterative way, all-order SD+vT approximation.

Some of the ideas and results presented in Chapters 4, 5, and 6 have been published and are subject to copyright laws. These publications are cited accordingly.