Gamma-ray halos around pulsars: impact on pulsar wind physics and galactic cosmic ray transport

TeV haloes are a recently discovered class of very high energy gamma-ray emitters. These sources consist of extended regions of multi-TeV emission, originally observed around the two well-known and nearby pulsars, Geminga and PSR B0656+14 (Monogem), and possibly, with different degrees of confidence, around few more objects with similar age. Since their discovery, TeV haloes have raised much interest in a large part of the scientific community, for the implications their presence can have on a broad range of topics spanning from pulsar physics to cosmic ray physics and dark matter indirect searches. In this article, we review the reasons of interest for TeV haloes and the current status of observations. We discuss the proposed theoretical models and their implications, and conclude with an overlook on the prospects for better understanding this phenomenon.