This review paper describes the overall operating principle of a discrete-time delta-sigma modulator (DTDSM) and a continuous-time delta-sigma modulator (CTDSM) using a switched-capacitor (SC). In addition, research that has solved the problems related to each delta-sigma modulator (DSM) is introduced, and the latest developments are explained. This paper describes the chopper-stabilization technique that mitigates flicker noise, which is crucial for the DSM. In the case of DTDSM, this paper addresses the problems that arise when using SC circuits and explains the importance of the operational transconductance amplifier performance of the first integrator of the DSM. In the case of CTDSM, research that has reduced power consumption, and addresses the problems of clock jitter and excess loop delay is described. The recent developments of the analog front end, which have become important due to the increasing use of wireless sensors, is also described. In addition, this paper presents the advantages and disadvantages of the three-opamp instrumentation amplifier (IA), current feedback IA (CFIA), resistive feedback IA, and capacitively coupled IA (CCIA) methods for implementing instrumentation amplifiers in AFEs.