More than 140 different molecules have been identified in the interstellar medium (ISM) to date. Dust grain particles are also found in the ISM, and some of these molecules freeze out at the cold temperatures (10–20 K) to form molecular ices. Understanding the adsorption and desorption of these ices is crucially important in understanding the processes that lead to star and planet formation, and may even help to lead to an understanding of the origin of life itself. High sensitivity surface science techniques, including temperature programmed desorption (TPD) and reflection absorption infrared spectroscopy (RAIRS), are being increasingly used to investigate the interactions between dust grains and interstellar ices. This perspective provides an overview of the current level of understanding of the adsorption and desorption of astrophysically relevant molecules from a range of dust grain analogue surfaces. Whilst the focus of this review is on interstellar ices, the results discussed are equally valid to discussions of cometary and planetary ices.