Hybrid systems of laser-cooled trapped ions and ultracold atoms combined in a\nsingle experimental setup have recently emerged as a new platform for\nfundamental research in quantum physics. This paper reviews the theoretical and\nexperimental progress in research on cold hybrid ion-atom systems which aim to\ncombine the best features of the two well-established fields. We provide a\nbroad overview of the theoretical description of ion-atom mixtures and their\napplications, and report on advances in experiments with ions trapped in Paul\nor dipole traps overlapped with a cloud of cold atoms, and with ions directly\nproduced in a Bose-Einstein condensate. We start with microscopic models\ndescribing the electronic structure, interactions, and collisional physics of\nion-atom systems at low and ultralow temperatures, including radiative and\nnon-radiative charge transfer processes and their control with magnetically\ntunable Feshbach resonances. Then we describe the relevant experimental\ntechniques and the intrinsic properties of hybrid systems. In particular, we\ndiscuss the impact of the micromotion of ions in Paul traps on ion-atom hybrid\nsystems. Next, we review recent proposals for using ions immersed in ultracold\ngases for studying cold collisions, chemistry, many-body physics, quantum\nsimulation, and quantum computation and their experimental realizations. In the\nlast part we focus on the formation of molecular ions via spontaneous radiative\nassociation, photoassociation, magnetoassociation, and sympathetic cooling. We\ndiscuss applications and prospects of cold molecular ions for cold controlled\nchemistry and precision spectroscopy.\n