Medical ultrasound has become a crucial part of modern society and continues to play a vital role in the diagnosis and treatment of illnesses. Over the past decades, the development of medical ultrasound has seen extraordinary progress as a result of the tremendous research advances in microelectronics, transducer technology and signal processing algorithms. However, medical ultrasound still faces many challenges including power-efficient driving of transducers, low-noise recording of ultrasound echoes, effective beamforming in a non-linear, high-attenuation medium (human tissues) and reduced overall form factor. This paper provides a comprehensive review of the design of integrated circuits for medical ultrasound applications. The most important and ubiquitous modules in a medical ultrasound system are addressed, i) transducer driving circuit, ii) low-noise amplifier, iii) beamforming circuit and iv) analog-digital converter. Within each ultrasound module, some representative research highlights are described followed by a comparison of the state-of-the-art. This paper concludes with a discussion and recommendations for future research directions.