Over the last decade there has been rapid progress in studying ultracold mixtures of trapped ions and neutral atoms. These systems are particularly interesting for studying quantum impurity physics and quantum chemistry, and may find applications in buffer gas cooling of trapped ions. In these lecture notes I review some of the recent progress and give relevant background information that may be useful for students who are considering working in this research area, but already have a firm knowledge of quantum and atomic physics. The notes are written from the Amsterdam perspective and therefore are not meant as a comprehensive review. I introduce the interaction potential between atoms and ions as well as some relevant atomic physics background. Then I explain how ion traps work. I treat the problem of a trapped ion in a neutral buffer gas and under what circumstances the gas can cool the ion. Next, I explain how the motional state of trapped ions can be measured. For applications in quantum technology, it will be of vital importance to be able to control the interactions between ion and atoms. I treat the case of using Rydberg dressing or excitation to boost the range and strength of the atom-ion interactions. Finally, I treat a number of applications in quantum chemistry that were studied in the group in Amsterdam, followed by a brief outlook.