Siege
The optimal result of most standoffs is a peaceful resolution: the safe extraction of hostages and bystanders, and the peaceful surrender and arrest of the hostage-takers. To ensure this, police make use of trained negotiators and psychologists to learn the hostage-takers' demands (and meet said demands if feasible or permissible), gain the hostage-takers' trust, clarify that police do not intend to kill them or will even let them go (regardless of whether such claims are true), and coax the hostage-takers into surrendering or at least releasing hostages. In the event a peaceful resolution is impossible—negotiations fail or do not proceed, hostages are released but the hostage-takers refuse to surrender, the hostage-takers resist violently, or hostages are killed—police may respond in force, generally being able to rely on police tactical units or even military support if possible and required.

Most standoffs are much shorter than military sieges, often lasting hours or days at most. Lengthy sieges may still occur, albeit rarely, such as the 51-day-long 1993 Waco siege. Most standoffs end in a peaceful resolution (i.e. 1973 Brooklyn hostage crisis), though some may end in a police or military assault (i.e. 1994 Air France Flight 8969 hijacking, 1980 Iranian Embassy siege) or, in the worst-case scenarios, the deaths of authorities, hostage-takers, or hostages (i.e. 1985 MOVE bombing, 1985 EgyptAir Flight 648 hijacking, 2004 Beslan school siege, 2022 Robb Elementary School shooting). The aforementioned worst-case scenarios often result from poor planning, tactics, or negotiations on the part of the authorities (e.g. accidental killings of hostages by Unit 777 during the EgyptAir Flight 648 hijacking), or from violent acts committed by the hostage-takers (e.g. suicide bombings and executions during the Beslan school siege).
In some jurisdictions, depending on certain circumstances, standoffs that would usually be handled by police may be transferred to the military. For example, in the United Kingdom, standoffs with terrorists may be transferred to military responsibility for a military assault on the besieged. The threat of such an action ended the 1975 Balcombe Street siege, but the 1980 Iranian Embassy siege ended in a military assault and the deaths of all but one of the hostage-takers.

See also: Battleplan (documentary TV series)
Blitzkrieg
Breastwork (fortification)
Infiltration
Last stand
Maneuver warfare
Medieval warfare
Sangar (fortification)
Siege engines
Siege equipment
Tunnel warfare
Lists

List of established military terms
List of sieges

Notes: 

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Needham, Joseph (1986). Science and Civilization in China. Vol. 5. Taipei: Caves Books Ltd. 
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Turnbull, Stephen R. (2002). Siege Weapons of the Far East. Oxford: Osprey Publishing Ltd.
Wheelis, M. (2002). "Biological warfare at the 1346 siege of Caffa". Emerg Infect Dis. 8 (9). Center for Disease Control: 971–975. doi:10.3201/eid0809.010536. PMC 2732530. PMID 12194776.
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Further reading: Duffy, Christopher (1996) [1975]. Fire & Stone: The Science of Fortress Warfare (1660–1860) (2nd ed.). New York: Stackpole Books.
Duffy, Christopher (1996). Siege Warfare: Fortress in the Early Modern World, 1494–1660. Routledge and Kegan Paul.
Duffy, Christopher (1985). Siege Warfare, Volume II: The Fortress in the Age of Vauban and Frederick the Great. London: Routledge and Kegan Paul.
Garlan, Yvon (1974). Recherches de poliorcétique grecque (in French). Paris: De Boccard.
Lynn, John A. (1999). The Wars of Louis XIV. Pearson. ISBN 0582056292.
May, Timothy. (2004). "Mongol Arms". Explorations in Empire, Pre-Modern Imperialism Tutorial: the Mongols. University of Wisconsin-Madison. Archived from the original on 18 May 2008.
Ostwald, Jamel (2007). Vauban Under Siege: Engineering Efficiency and Martial Vigor in the War of the Spanish Succession. History of Warfare. Vol. 41 (illustrated ed.). Brill. ISBN 978-90-04-15489-6.
Warner, Philip (1968). Sieges of the Middle Ages. G. Bell & Sons.
Historiography

Bachrach, Bernard S (1994). "Medieval siege warfare: a reconnaissance". Journal of Military History. 58 (1): 119–133. doi:10.2307/2944182. JSTOR 2944182.

External links: 
Native American Siege Warfare.