The Battle of France began on May 10, 1940 when the Luftwaffe launched a massive attack in the early hours on airfields in the Netherlands, Belgium, and France, trying to destroy as many planes on the ground. Thousands of German paratroopers jumped above Holland and Belgium, occupying strategic points.
The Allies had correctly anticipated that a direct assault on the Maginot Line was unlikely and that the Germans would rather attack through Belgium and the Netherlands; therefore the first German movement did not surprise them.
After the Wehrmacht had begun the offensive on the Low Countries, the most mobile and well-trained French units and the British Expeditionary Force (BEF) crossed the Franco-Belgian border. The Allies intended to form a defensive line that ran from the Dutch city of Breda, continued along the river Dyle through Belgium, and reached up to the Maginot Line.
Derivative work by Tazadeperla [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0/deed.en)], via Wikimedia Commons
The use of airborne troops by the Germans had provided them quickly not only the capture of strategic bridges but also apparently impregnable strongholds, as was the Belgian Fort Eben-Emael.
One of the biggest tank battles in the entire campaign took place in Belgium, near Hannut, when two French cavalry armored divisions faced two German armored divisions. Although the high quality of the French Somua S35 tanks caused the loss of many German tanks, the Wehrmacht's envelopment tactics eventually compelled the French to withdraw.
However, the major surprise of the Battle of France came on May 12, when no less than seven of the ten Panzer divisions which the Germans had, attacked through the hilly wooded region of the Ardennes, forcing the crossing of the River Meuse between Dinant and Sedan. In that area, the French had divisions consisting mainly of reservists who succumbed in the face of German assault, after Junker Ju-87 Stuka dive bombers released powerful air raids on them. Both the Allied aircraft attacks and the counterattack of the French 3rd Armored Division failed, the Germans consolidating their bridgehead.
Bundesarchiv, Bild 121-0412 / [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0/de/deed.en)], via Wikimedia Commons
Then, while some of his forces were guarding the southern flank, the German General Heinz Guderian launched the famous "sickle-cut", advancing towards the English Channel with two Panzer divisions. A little to the north, Erwin Rommel, commander of the 7th Panzer Division, proceeded in the same direction with his unit.
The Allies tried to impede the panzers march towards the English Channel, first by the French 4th Armored Division, led by Colonel Charles de Gaulle, which attacked twice the divisions of Guderian, but the Germans repelled them with the contribution of the Luftwaffe.
The British attack at Arras, on 21 May, with Matilda tanks formidable armored for the year 1940, started well but was ultimately defeated by the Germans with the help of heavy artillery and the dreaded 88mm Flaks.
Managing to defeat the Allied counterattacks, the Germans reached the English Channel. Luckily for the British, for several days, only the Luftwaffe tried to block the BEF evacuation through the port of Dunkirk. Thus Britain managed to evacuate its soldiers in Operation Dynamo, as well as a substantial number of French soldiers.
Without allies and with their best forces lost, the French still organized a new line of defense, from the river Somme continuing with the Maginot Line. When the panzer divisions attacked again on June 5, generally bypassed the concentric defensive areas called "hedgehogs" and penetrated in depth, leaving them to be annihilated by infantry divisions which were coming from behind.
The German air supremacy proved decisive in the outcome of Battle of France. On June 14, Paris fell while the German tanks were advancing behind the Maginot Line to the Swiss border. Marshal Phillippe Petain asked for a truce, which resulted in a German occupation of the northern and western France, while the south of the country remained administrated by a French government obedient to Berlin.