Most of the ordinary matter in the local Universe has not been converted into stars but resides in a largely unexplored diffuse, hot, X-ray emitting plasma. It pervades the gravitational potentials of massive galaxies, groups and clusters of galaxies, as well as the filaments of the cosmic web. The physics of this hot medium, such as its dynamics, thermodynamics and chemical composition can be studied using X-ray spectroscopy in great detail. Here, we present an overview of the basic properties and discuss the self similarity of the hot "atmospheres" permeating the gravitational halos from the scale of galaxies, through groups, to massive clusters. Hot atmospheres are stabilised by the activity of supermassive black holes and, in many ways, they are of key importance for the evolution of their host galaxies. The hot plasma has been significantly enriched in heavy elements by supernovae during the period of maximum star formation activity, probably more than 10 billion years ago. High resolution X-ray spectroscopy just started to be able to probe the dynamics of atmospheric gas and future space observatories will determine the properties of the currently unseen hot diffuse medium throughout the cosmic web.

Accepted for publication in the book "Reviews in Frontiers of Modern Astrophysics: From Space Debris to Cosmology" (eds Kabath, Jones and Skarka; publisher Springer Nature) funded by the European Union Erasmus+ Strategic Partnership grant "Per Aspera Ad Astra Simul" 2017-1-CZ01-KA203-035562