The development of lensfree on-chip microscopy in the past decade has opened up various new possibilities for biomedical imaging across ultra-large fields of view using compact, portable, and cost-effective devices. However, until recently, its ability to resolve fine features and detect ultra-small particles has not rivalled the capabilities of the more expensive and bulky laboratory-grade optical microscopes. In this Frontier Review, we highlight the developments over the last two years that have enabled computational lensfree holographic on-chip microscopy to compete with and, in some cases, surpass conventional bright-field microscopy in its ability to image nano-scale objects across large fields of view, yielding giga-pixel phase and amplitude images. Lensfree microscopy has now achieved a numerical aperture as high as 0.92, with a spatial resolution as small as 225 nm across a large field of view e.g., >20 mm(2). Furthermore, the combination of lensfree microscopy with self-assembled nanolenses, forming nano-catenoid minimal surfaces around individual nanoparticles has boosted the image contrast to levels high enough to permit bright-field imaging of individual particles smaller than 100 nm. These capabilities support a number of new applications, including, for example, the detection and sizing of individual virus particles using field-portable computational on-chip microscopes.