Optical compound microscope has been a major tool in biomedical imaging for centuries. Its performance relies on relatively complicated, bulky and expensive lenses and alignment mechanics. In contrast, the lensless microscope digitally reconstructs microscopic images of specimens without using any lenses, as a result of which it can be made much smaller, lighter and lower-cost. Furthermore, the limited space-bandwidth product of objective lenses in a conventional microscope can be significantly surpassed by a lensless microscope. Such lensless imaging designs have enabled high-resolution and high-throughput imaging of specimens using compact, portable and cost-effective devices to potentially address various point-of-care, global-health and telemedicine related challenges. In this review, we discuss the operation principles and the methods behind lensless digital holographic on-chip microscopy. We also go over various applications that are enabled by cost-effective and compact implementations of lensless microscopy, including some recent work on air quality monitoring, which utilized machine learning for high-throughput and accurate quantification of particulate matter in air. Finally, we conclude with a brief future outlook of this computational imaging technology.