This paper provides a comprehensive review of compressed sensing or compressive sampling (CS) in bioelectric signal compression applications. The aim is to provide a detailed analysis of the current trends in CS, focusing on the advantages and disadvantages in compressing different biosignals and its suitability for deployment in embedded hardware. Performance metrics such as percent root-mean-squared difference (PRD), signal-to-noise ratio (SNR), and power consumption are used to objectively quantify the capabilities of CS. Furthermore, CS is compared to state-of-the-art compression algorithms in compressing electrocardiogram (ECG) and electroencephalography (EEG) as examples of typical biosignals. The main technical challenges associated with CS are discussed along with the predicted future trends.