In dense media positrons and positronium atoms are bound in self-trapped states. These states are density fluctuations stabilised by the light quantum particles. The positronium atoms are trapped in a bubble, while positrons are trapped in a cluster. Transitions to these states at changes of the density or temperature of the medium produce an essential effect on the annihilation rate. It can be considered as a local phase transition in the vicinity of the positron (or positronium). The present article is a review of the present state of the physics of positron and positronium self-trapped states in dense gases and liquids. Basic properties of the positronium bubbles and positron clusters are considered, as well as the conditions under which they exist and their effect on the annihilation rate. Experiments in which the self-trapping was detected are discussed. The positron and positronium self-trapping is considered to be closely related to other similar electronic phenomena, known for dense gases, liquids and other dense media. The main results from the study of positron and positronium interactions with isolated atoms are also presented. The density effects in moderately dense gases anticipating self-trapping are considered.