The region of space between stars, the interstellar medium, has been found to contain over 160 chemical species to date. These molecules are contained within regions of gas and dust, measuring several light years across, known as interstellar dust clouds. Many of these molecular species are formed in the gas phase, for example, via the reactions of molecules with ions. However, some critical gas phase processes are often slow due to the low temperatures and pressures found in the interstellar medium and cannot readily account for the abundances of some species. Consequently reactions on the surfaces of interstellar dust grains are often invoked to explain the abundances some molecules. These dust grains represent approximately 1 % of the mass of a typical interstellar dust cloud and typically consist of carbon, silicates or metal oxides. The temperature of these interstellar dust grains is low enough (~ 10 K) that over time icy mantles consisting of simple atomic and molecular species can build up on their surfaces. Whether and how these simple species can be processed to form more complex molecules such as alcohols, simple sugars and potentially amino acids is a key astrochemical problem. One way in which astrophysical ices can be processed to form more complex species is via the reactions of species within the ice with simple free radicals such as H, C, N and O. This thesis therefore presents experimental studies of the reactions of atomic species with some astrophysically relevant molecular ices under interstellar conditions. Since hydrogen and oxygen are the first and third most abundant elements in the interstellar medium respectively, these experiments have specifically focussed upon the reactions of hydrogen and oxygen atoms. In addition to the characterization of surface reactions between key astrochemical species, kinetic parameters for use in astrochemical models are derived from these experiments.