Optical tweezers and associated manipulation tools in the far field have had a major impact on scientific and engineering research by offering precise manipulation of small objects. More recently, the possibility of performing manipulation with surface plasmons has opened opportunities not feasible with conventional far-field optical methods. The use of surface plasmon techniques enables excitation of hotspots much smaller than the free-space wavelength; with this confinement, the plasmonic field facilitates trapping of various nanostructures and materials with higher precision. The successful manipulation of small particles has fostered numerous and expanding applications. In this paper, we review the principles of and developments in plasmonic tweezers techniques, including both nanostructure-assisted platforms and structureless systems. Construction methods and evaluation criteria of the techniques are presented, aiming to provide a guide for the design and optimization of the systems. The most common novel applications of plasmonic tweezers, namely, sorting and transport, sensing and imaging, and especially those in a biological context, are critically discussed. Finally, we consider the future of the development and new potential applications of this technique and discuss prospects for its impact on science.