Space warfare
This option tends to be highly destructive and indiscriminate as any attack will produce space debris, which can indiscriminately affect other satellites in similar orbits. While this option comes with the benefit of leveraging existing technologies and a certain element of surprise, as an attack cannot be detected until a missile has exited its silo, there are significant downsides. Firstly, there is the cost disparity of using an ICBM or ABM to kill a small and inexpensive satellite. Additionally, these missiles are not designed to send payloads out to geocentric orbit, as such they can only affect targets in low earth orbit and only in a target area centered around the static location of the missile itself.

Co-Orbital: Co-orbital systems come with a few potential kill mechanisms: in guided kinetic vehicles, like the Multiple Kill Vehicle, or in the form of a satellite that can release a kinetic interceptor or a cloud of debris. The first co-orbital system, Istrebitel Sputnikov, was developed by the Soviet Union in the 1970s and reportedly utilized one of these mechanisms.
There are allegations that Russia continues to test co-orbital ASAT weapons as recently as 2020. In 2020 the U.S. State Department claimed that a Russian satellite, Cosmos-2519, exhibited behavior “inconsistent” with its intended mission. While in orbit, Kosmos-2519 deployed a smaller satellite, which Russian state media claimed: “conducted autonomous flight, a change in orbit, and a satellite inspection before returning to the base station”. Another incident back in 2019 involved two Russian satellites, Kosmos 2542 and 2543, one of which appeared to begin following a U.S. national security satellite. Such “inspector” satellites can be armed with lasers to provide non-destructive interference or deadly kinetic interceptors.
While these co-orbital systems provide more utility when compared to more direct and destructive options, their advantages are contingent on being maneuverable and inconspicuous. Given the increasing paranoia surrounding co-orbital anti-satellite, it is hard to believe that the major players in space will fail to notice the deployment of “research” satellites.

Space warfare in science fiction: Space warfare is a staple of science fiction, where it is shown with a wide range of realism and plausibility. Fictional space warfare includes anticipated future technology and tactics, and fantasy- or history-based scenarios in a scifi setting. Some portray a space military as like an air force; others depict a more naval framework. Still others suggest forces more like space marine: highly mobile forces doing interplanetary and interstellar war but most of the conflict happens in terrestrial environments. The main sub-genres of the Space warfare in science fiction thematic genre are space opera, Military and Space Western. Though sword and planet stories like Finisterre universe by C. J. Cherryh might be considered, they rarely feature such technologies. These three genres often intertwine and have themes that are common to all. Written Space Westerns are often based directly on existing established scifi space opera franchises with expanded universes like Star Wars and Star Trek, including Warhammer 40,000: the most popular space opera military miniature wargame which spawned successful spin-off media: novels, video-games and on-going live adaption based on books by Dan Abnett.
Both kinetic and directed energy weapons are often seen, along with various military space vessels. E. E. Smith's Lensman is an early example, which also inspired the term space opera due to the grandiose scale of the stories. Orson Scott Card's Ender's Game series is a notable example in that it makes a conjecture as to what sort of tactics and training would be needed for war in outer space. Other scifi authors have also delved into the tactics of space combat, such as David Weber in his Honorverse series, and Larry Niven and Jerry Pournelle in their Mote in God's Eye series. A more recent example is Alastair Reynolds' Revelation Space universe, which explores combat at relativistic speed. Robert A. Heinlein's Starship Troopers is perhaps one of the best-known and earliest explorations of the "space marine" idea.
Space-based vehicular combat is portrayed in many movies and video games, most notably Star Wars, Stargate, Halo, Descent, Gundam, Macross, Babylon 5, and Star Trek. Games such as the Homeworld series have interesting concepts for space warfare, such as 3D battle formations, plasma-based projectors that get their energy from a ship's propulsion system, and automated uncrewed space combat vehicles. Other series, such as Gundam, prominently show vehicular combat in and among many near future concepts, such as O'Neill cylinders.
Fictional galaxies with space warfare are far too many to list, but popular examples include Star Trek (in all of its forms), Star Wars, Halo, Stargate, Warhammer 40,000, Babylon 5, Buck Rogers, Flash Gordon, Battlestar Galactica, Mass Effect, Freespace and many comic book franchises. Video games often touch the subject; the Wing Commander franchise is a prototypical example. Few games try to simulate realistic distance and speed, though Independence War and Frontier: Elite II both do, as does the board game Attack Vector: Tactical.
Many authors have either used a galaxy-spanning fictional empire as background or written about the growth and/or decline of such an empire. Said empire's capital is often a core world, such as a planet relatively close to a galaxy's supermassive black hole. Characterization can vary wildly from malevolent forces attacking sympathetic victims to apathetic bureaucracies to more reasonable entities focused on social progress, and anywhere in between. Scifi writers generally posit some form of faster-than-light drive in order to facilitate interstellar war. Writers such as Larry Niven have developed plausible interplanetary conflict based on human colonization of the asteroid belt and outer planets via technologies using currently known physics.

See also: Asteroid impact avoidance
Beijing–Washington space hotline
Militarisation of space
Space force
Space weapon
Starlink satellite services in Ukraine, instance of a satellite constellation being used for warfare
Sun outage
Related to specific countries and facilities:

Department of Defense Manned Space Flight Support Office
European Aeronautic Defense and Space Company
Joint Functional Component Command for Space and Global Strike (US Strategic Command)
National Missile Defense
Pine Gap (Australia)
United States Air Force Space Command
United States Army Space and Missile Defense Command

References: 

Further reading: Hobbes, D (1986): An Illustrated Guide to Space Warfare Salamander Books Ltd. ISBN 0-86101-204-6.
Macvey, John W.: Space Weapons, Space War. New York: 1979 Stein and Day (written by a professional astronomer). ISBN 978-0812861112.
David Jordan: Air and Space Warfare, pp. 178–223, in:Understanding modern warfare. Cambridge Univ. Press, Cambridge 2008, ISBN 978-0-521-87698-8.
John J. Klein: Space Warfare: Strategy, Principles and Policy. Routledge, Oxford 2006, ISBN 978-0-415-40796-0.
Joan Johnson-Freese: Space Warfare in the 21st Century – Arming the Heavens. Routledge, Oxford 2016,  ISBN 978-1-138-69388-3.