Additive manufacturing, also called 3D printing, is a rapidly evolving technique that allows for the fabrication of functional materials with complex architectures, controlled microstructures, and material combinations. This capability has influenced the field of biomedical sensing devices by enabling the trends of device miniaturization, customization, and elasticity (i.e., having mechanical properties that match with the biological tissue). In this paper, the current state-of-the-art knowledge of biomedical sensors with the unique and unusual properties enabled by 3D printing is reviewed. The review encompasses clinically important areas involving the quantification of biomarkers (neurotransmitters, metabolites, and proteins), soft and implantable sensors, microfluidic biosensors, and wearable haptic sensors. In addition, the rapid sensing of pathogens and pathogen biomarkers enabled by 3D printing, an area of significant interest considering the recent worldwide pandemic caused by the novel coronavirus, is also discussed. It is also described how 3D printing enables critical sensor advantages including lower limit-of-detection, sensitivity, greater sensing range, and the ability for point-of-care diagnostics. Further, manufacturing itself benefits from 3D printing via rapid prototyping, improved resolution, and lower cost. This review provides researchers in academia and industry a comprehensive summary of the novel possibilities opened by the progress in 3D printing technology for a variety of biomedical applications.