The majority of microelectromechanical system (MEMS) devices must be combined with integrated circuits (ICs) for operation in larger electronic systems. While MEMS transducers sense or control physical, optical or chemical quantities, ICs typically provide functionalities related to the signals of these transducers, such as analog-to-digital conversion, amplification, filtering and information processing as well as communication between the MEMS transducer and the outside world. Thus, the vast majority of commercial MEMS products, such as accelerometers, gyroscopes and micro-mirror arrays, are integrated and packaged together with ICs. There are a variety of possible methods of integrating and packaging MEMS and IC components, and the technology of choice strongly depends on the device, the field of application and the commercial requirements. In this review paper, traditional as well as innovative and emerging approaches to MEMS and IC integration are reviewed. These include approaches based on the hybrid integration of multiple chips (multi-chip solutions) as well as system-on-chip solutions based on wafer-level monolithic integration and heterogeneous integration techniques. These are important technological building blocks for the 'More-ThanMoore' paradigm described in the International Technology Roadmap for Semiconductors. In this paper, the various approaches are categorized in a coherent manner, their merits are discussed, and suitable application areas and implementations are critically investigated. The implications of the different MEMS and IC integration approaches for packaging, testing and final system costs are reviewed.