Maneuver warfare
The results of that review were seen in the Franco-Prussian War.

Mechanization: In the mid-19th century, various forms of mechanized transport were introduced, starting with trains running on steam power. That resulted in  significant logistic improvements. Opposing armies were no longer limited in speed by the pace of march. Some train-borne maneuvering took place during the American Civil War in the 1860s, but the sizes of the armies involved meant that the system could provide only limited support. Armored trains were among the first armored fighting vehicles employed by mankind.
During the Franco-Prussian War, the Prussians, knowing that the French could field a larger army than theirs, made a plan that required speed by surrounding the French strongpoints and destroying or bypassing them; it was called the Kesselschlacht, or 'cauldron battle'. The remainder of the army could advance unopposed to take important objectives. If war was declared, Prussia could quickly mobilize and invade, destroy French field forces, and win before the French army could fully react. That tactic was used to devastating effect in 1870 since Prussian forces surrounded and defeated French forces, captured Napoleon III and besieged Paris. The Germans' battle plans for World War I were similar. Germany attempted to repeat the "knock-out blow" against the French armies in the Schlieffen Plan. However, technology evolved significantly in the preceding four decades; both the machine gun and more powerful artillery shifted the balance of power toward the defense. All combatants were desperate to get the front moving again, but that proved to be difficult.
Germany introduced new tactics with infiltration and stormtrooper "shock troops" toward the end of World War I to bypass resistance. Russian general Aleksei Brusilov used similar tactics in 1916 on the Eastern Front during the Brusilov Offensive.
The introduction of fully armored tanks, in a series of increasingly successful operations, presented a way out of the deadlock of attrition and trench warfare, but World War I ended before the British would field thousands of tanks to be put in a large-scale offense. Fuller had proposed Plan 1919 to use tanks to break through the lines and then to wreak havoc on the German lines of supply and communication.
During the interwar period, the British developed ideas for fully-mechanized all-arms warfare with the Experimental Mechanized Force. The Germans reviewed their doctrine and revised their approach by expanding on infiltration tactics and amplifying them with motor transport. Heinz Guderian was a leading proponent of armored combat. The German military stressed several key elements: versatile tanks combined with mobile infantry and artillery, close air support, rapid movement and concentration of forces (Schwerpunkt), and aggressive independent local initiative. All was strictly coordinated by radio and contributed to new tactics during the Battle of France in 1940. Theories in Germany about armored warfare have some similarities with interwar theories of British officers J.F.C. Fuller and B. H. Liddell Hart, which the British army failed to embrace and understand fully.
There are similarities between blitzkrieg and the Soviet concept of "deep battle," which the Soviets used to great effect in 1944 and continued to use as a doctrine during the Cold War.

Soviet deep battle: In the Soviet Union during the 1920s and the 1930s, the concept of "deep battle" was developed and integrated into the Red Army field regulations doctrine by Marshal Mikhail Tukhachevsky. That led to the creation of cavalry mechanised groups during World War II and to operational maneuver groups during the Cold War.

US Marine Corps doctrine: According to the US Marine Corps, one key concept of maneuver warfare is that maneuver is traditionally thought of as a spatial concept, the use of maneuver to gain positional advantage. The US Marine concept of maneuver, however, is a "warfighting philosophy that seeks to shatter the enemy's cohesion through a variety of rapid, focused, and unexpected actions which create a turbulent and rapidly deteriorating situation with which the enemy cannot cope."

The US Marine manual goes on to say: "This is not to imply that firepower is unimportant. On the contrary, firepower is central to maneuver warfare. Nor do we mean to imply that we will pass up the opportunity to physically destroy the enemy. We will concentrate fires and forces at decisive points to destroy enemy elements when the opportunity presents itself and when it fits our larger purposes."
The possibility of a massive Soviet offensive in Western Europe led to the creation of the US Army's AirLand battle doctrine. Though far from focusing on maneuver, it emphasized using combined arms to disrupt an enemy's plan by striking through their depth and was seen as moving toward maneuver warfare in comparison to the earlier active defense concept. The AirLand doctrine was seen by Martin van Creveld as "arguably a half way house between maneuver and attrition".

British air maneuver doctrine: The British Joint Forces are limited to consider air assault or airmobile operations in their 2016 publication "Joint Doctrine Note on Air Manoeuvre".

Limitations in a modern context: A key requirement for success in maneuver warfare is up-to-date accurate intelligence on the disposition of key enemy command, support, and combat units. In operations whose intelligence is either inaccurate, unavailable, or unreliable, the successful implementation of strategies based on maneuver warfare can become problematic. When faced with a maneuverable opponent capable of redeploying key forces quickly and discreetly or when tempered, the capacity of maneuver warfare strategies to deliver victory becomes more challenging.
The 2006 Lebanon War is an example of such shortcomings being exposed. Despite overwhelming firepower and complete air superiority, Israeli forces were unable to deliver a decisive blow to the command structure of Hezbollah or to degrade its effective capacity to operate. Although inflicting heavy damage, Israel was unable to locate and destroy Hezbollah's diluted force dispositions or to neutralize key command centers. Therefore, it did not meet its war aims. The insurgency in Iraq also demonstrates that a military victory over an opponent's conventional forces does not automatically translate into a political victory.
Some military theorists such as William Lind and Colonel Thomas X. Hammes propose to overcome the shortcomings of maneuver warfare with the concept of what they call fourth generation warfare. For example, Lieutenant-Colonel S.P. Myers writes that "maneuver is more a philosophical approach to campaign design and execution than an arrangement of tactical engagements". Myers goes on to write that maneuver warfare can evolve and that "maneuverist approach in campaign design and execution remains relevant and effective as a counter-insurgency strategy at the operational level in contemporary operations."
In the early stages of the 2022 Russian invasion of Ukraine, Russia's advances were stalled by Ukraine's widespread deployment of man-portable anti-tank missiles. The scholar Seth Jones argued that Russia was forced to abandon maneuver warfare after an inept failure to apply combined arms, forcing a transition to a war of attrition. In comparison, Ukrainian forces used maneuver warfare more effectively during the 2024 Kursk Oblast incursion while Russian forces struggled to repel the attack. According to Michael Kofman "Russian forces do far better when they’re operating with prepared defense, fixed lines, more on positional warfare."

See also: Charge
Decision cycle
Defeat in detail
Flanking maneuver
Historical examples of flanking maneuvers
OODA loop
Pincer movement
Russian military deception
Operational manoeuvre group

References: 

Sources: Boyd, John. Patterns of Conflict. 1986.
Simpkin, Richard E. Race to the Swift: Thoughts on Twenty-First Century Warfare. Brassey's, 2000.
Richard Simpkin in association with John Erickson  Deep Battle: The Brainchild of Marshal Tukhachevskii, London, Brassey's Defence, 1987.  ISBN 0-08-031193-8
Lind, William S. Maneuver Warfare Handbook. 1985. Westview Special Studies in Military Affairs. Westview Press Inc. Boulder, CO.
Leonhard, Robert.The Art of Maneuver: Maneuver-Warfare Theory and Air-Land Battle.  1991. Presidio Press. Novato, CA.
MajGen Ray Smith, USMC (ret.) and Bing West (1 August 2003). "Implications from Iraqi Freedom for the Marine Corps". Westwrite. Archived from the original on 27 September 2007. Retrieved 19 May 2007. The first war fought under the new doctrine of Maneuver Warfare, with several observations and suggestions for future change.