The ability to manipulate and sense biological molecules is important in many life science domains, such as single-molecule biophysics, the development of new drugs and cancer detection. Although the manipulation of biological matter at the nanoscale continues to be a challenge, several types of nanotweezers based on different technologies have recently been demonstrated to address this challenge. In particular, photonic and plasmonic nanotweezers are attracting a strong research effort especially because they are efficient and stable, they offer fast response time, and avoid any direct physical contact with the target object to be trapped, thus preventing its disruption or damage. In this paper, we critically review photonic and plasmonic resonant technologies for biomolecule trapping, manipulation, and sensing at the nanoscale, with a special emphasis on hybrid photonic/plasmonic nanodevices allowing a very strong light-matter interaction. The state-of-the-art of competing technologies, e.g., electronic, magnetic, acoustic and carbon nanotube-based nanotweezers, and a description of their applications are also included.