A blitzkrieg is an operational level military doctrine which calls for a quick surprise attack by massive ground and air forces. "Blitzkrieg" is from the German words for lighting ("blitz") and war ("krieg"). The term was not used by German military planners, originating at the Propaganda Ministry of Josef Goebbels. It was used in press accounts to describe the lightning speed of the German military during World War II. Modern American "Shock and Awe" doctrine is based upon the lessons of blitzkrieg.
Basic Requirements and Features
Blitzkrieg or maneuver is based on the idea of "gaps and surfaces" or as the Germans called it "Flaechen und Luekentaktik", which means the tactics of surfaces and gaps. The basic intent of blitzkrieg is to avoid those points which are "surfaces" or hard points in the enemy line and target "gaps" or weakly defended areas. This was further developed by Basil Liddell Hart's concept of expanding torrent system tactic. The idea with this was to act like water: You poke and prod at the enemy's lines and, when a weak point is found, attack it and expand the breach in the line, just as flowing water would do to a weak dam, thus causing a massive flow through it. In other words, rather than throwing strength against strength as in attrition-oriented forms of combat, you focus on throwing your strength against the enemy's weakness.
This requires what is known as recon pull, or another way of saying it is have your reconnaissance assets find the gaps and then they exploit them, bringing more and more assets with them as they go. This is different from more conventional forms of attack where it is focused on the commander of the formations essentially pushing units forward, whether it be a gap or a surface in the enemy lines.
If suitable gaps cannot be found, they are created via the use of "Stosstruppentaktik" or shock troop tactics. This is when a combination of suppression (via artillery or other assets) assault (with infantry and armor) and exploitation (mainly with armor) is used. This is essentially punching a hole in the enemy lines at a desirable point. You suppress the enemy in order to keep his head down and prevent him from engaging and have the assault happen as soon as is possible after the suppressive fire is finished. Once this is done, the committing of what is generally a large reserve happens once the initial assault is successful.
Another way of forming gaps was via the use of support attacks which could force the enemy to redeploy his assets thus creating weak points in his lines that can be exploited.
Those points that are considered strong points or surfaces are generally bypassed and allowed to weaken due to it being cut off from its support. In this way blitzkrieg also dislocates the enemy in space.
Because blitzkrieg is more fluid in terms of maneuver and as such the speed with which situations change, it requires commanders to be close to the action rather than being behind the lines as what the stereotype was of generals from WW1. Guderian and Patton were known for this lead from the front leadership style. The reason that leaders must be up front is so that they can be able to provide the needed leadership at the decisive time at the decisive point. This allows for faster decision making and makes it so that the enemy is dislocated in time as well as space. This would be described as getting inside of your opponent's decision cycle for you are able to make it so that the opponent is reacting to you, if this continues there is the risk that his actions will be rendered obsolescent and as such paralyzes his ability to react to you.
Another requirement is to allow subordinates the ability to act independently based on what the commander's intent is. This allows for decisions made on the fly and helps to speed up the decision cycle. This is called "auftragstaktik" or mission tactics. It requires a combination of trust within the command structure and common ideas and experiences in terms of tactics and training for this to be successful. A commander must trust that his subordinates not only understand his intent but also must trust that they will be accurate in reporting the situations that they are facing on the front.
One of the risks of engaging in this style of warfare is that what you perceive as a gap may actually be a surface. If this happens the commander may run into what is known as a "fire sack" or a large scale ambush.
Before World War II
Colonel John Fuller, the chief of staff of the British Tank Corps, is credited with developing this tactic after his extreme disappointment with the effect of tanks during World War I. His ideas, which were ignored by the British Army, called for long-range mass tank attacks. He envisioned this attack to be supported by air power as well as motorized infantry and artillery support. He wrote two books in the 1920s which detailed his ideas: Reformation of War and Foundation of the Science of War. In 1926, the German Army began ordering tanks and vehicles which would allow them to implement Colonel Fuller's ideas. 
In 1937 Heinz Guderian wrote the book Achtung - Panzer! describing the combined arms tactics used initially by the Germans in WWII and later on by armies throughout the world. The Germans tested an application of these tactics in 1938 during the Spanish Civil War and as a result German High Command formulated plans to dismantle Polish defenses in 1939. 
During World War II
Blitzkrieg was a maneuver tactic where armored columns would break through the enemy lines, supported by artillery and attack aircraft. Once the columns were through the line, they would not stop but continue into the enemy rear, disrupting their supply lines and cutting off units. The armored columns were followed by infantry units that would exploit the gaps and surround the isolated enemy units. The tactic worked most effectively in Europe against the French and the Netherlands, and in the beginning of the conflict against the Soviet Union. Its success was due to a combination of surprise and favorable terrain.
There was three flaws to the Blitzkrieg tactic. First, it was heavily dependent on good weather and good terrain. If the terrain was hilly, combat was in an urban area, or if the weather turned the ground muddy (as in the Soviet Union), the armored units would slow down and become vulnerable to anti-tank units or worse to the enemy being able to react to attacks more effectively.
Second, air support was vital. Without it, the tanks were vulnerable to counter-attacks from enemy aircraft. This is true with all operations that focus heavily on the use of armored fighting vehicles, whether it be maneuver oriented or more attrition oriented. As the war progressed and the Germans lost control of the air, their panzer units suffered heavy losses from Allied air-to-ground aircraft such as the Soviet IL-2 and American P-47 Thunderbolt. German success in the Battle of the Bulge depended in part on the existence of cloud cover that would keep Allied air forces grounded.
Third, the blitzkrieg could be broken up by channeling the attack (holding the flanks) and attacking the point where the infantry and armored units met. This would separate the two and expose the armor to counter-attacks. The French and British used this tactic at the Battle of Arras in 1940, but lacked the armored units required to exploit their limited success.
The Soviets used all the above and had the additional fortune of vast swaths of land to design a defense-in-depth. As the panzer units moved farther and farther from their supplies, gaps would appear in their columns. These could then be attacked to break up the armor. Also, the Soviets used minefields to further disrupt the German armor.
By 1944, blitzkrieg as a German tactic was finished. The last actual appearance of blitzkrieg in the war when several German Armies smashed through the weak Allied lines in the Ardennes forest in what became known as the Battle of the Bulge. Luckily, their momentum did not hold, which marked an end to blitzkrieg warfare in WWII.
Blitzkrieg after the war
The first significant use of blitzkrieg tactics in the post-war era was in the 1967 Six Day War when Israel, after destroying almost the entire Egyptian air force on the ground in a little more than an hour, sent a three pronged tank attack south to seize the Sinai peninsula and north to occupy the Golan heights.
The ground phase of the first Gulf War, lasting only four days, followed a blitzkrieg-style attack with US, British, and French tanks that poured deep into southern Iraq in what General Norman Schwarzkopf termed a "Hail Mary" pass before hooking right and cutting off the escaping Iraqi forces, which were then demolished in detail by tanks and aircraft.
The resumption of hostilities in Iraq that led to the Iraq War relied upon "Shock and Awe" a derivative of blitzkrieg doctrine, but also a tactic used since the American Civil War, when massed artillery bombardment was first used to weaken and demoralize enemy forces before an advance.
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