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Analog integrated circuits (ICs) designed for processing physiological signals are important building blocks of wearable and implantable medical devices used for health monitoring or restoring lost body functions. Due to the nature of physiological signals and the corresponding application scenarios, the ICs designed for these applications should have low power consumption, low cutoff frequency, and low input-referred noise. In this paper, techniques for designing the analog front-end circuits with these three characteristics will be reviewed, including subthreshold circuits, bulk-driven MOSFETs, floating gate MOSFETs, and log-domain circuits to reduce power consumption; methods for designing fully integrated low cutoff frequency circuits; as well as chopper stabilization (CHS) and other techniques that can be used to achieve a high signal-to-noise performance. Novel applications using these techniques will also be discussed.