Singularities are ubiquitous in nature and generate high levels of interest and activity among scientists. At a singularity, a parameter of interest becomes indeterminate, a single point at which a mathematical quantity is not defined or not 'well behaved'. In optics, a singularity refers to a point at which some parameter describing the electromagnetic field becomes indeterminate. Even though at the singular point, things are not well defined, the neighbourhood points of a singularity are characterized by very large gradients and hence there is significant scope for new discoveries. Recent advances have gathered pace, and interest continues to grow in this field, and with its applications.Part of IOP Series in Advances in Optics, Photonics and Optoelectronics.