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Thousands of paratroops and glider-borne troops have also been dropped behind enemy lines and the Allies are already said to have penetrated several miles inland.
The landings were preceded by air attacks along the French coast.
About 1,300 RAF planes were involved in the first wave of assaults then 1,000 American bombers took up the attack dropping bombs on targets in northern France.
The Prime Minister Winston Churchill has told MPs that Operation Neptune - the codename for the Normandy landings - is proceeding "in a thoroughly satisfactory manner".
He said the landing of airborne troops was "on a scale far larger than anything there has been so far in the world" and had taken place with extremely little loss.
The assault began shortly after midnight under the command of General Bernard Montgomery.
Timing of the Normandy landings was crucial. They were originally scheduled to take place in May - then postponed until June and put off again at the last minute for 24 hours by bad weather.
Upwards of 4,000 ships and several thousand smaller craft crossed the Channel to the northern coast of France.
Enemy reports say the landings took place between the port of Le Havre and the naval base at Cherbourg.
King George VI broadcast a message last night warning of the "supreme test" the Allies faced and he called on the nation to pray for the liberation of Europe.
The Allied naval commander, Admiral Sir Bertram Ramsay, said the landings had taken the Germans completely by surprise. There were no enemy reconnaissance planes out and the opposition of coastal batteries was much less than expected.
He added: "There was a slight loss in ships but so slight that it did not affect putting armies ashore.
"We have got all the first wave of men through the defended beach zone and set for the land battle."
A statement broadcast from Berlin at midday said the German troops were "nowhere taken by surprise". It said many parachute units were wiped out on landing or taken prisoner.
Hits were also scored on battleships and on landing craft from the "guns of the Atlantic Wall" - the German defensive positions.
President Franklin D Roosevelt told a news conference the invasion did not mean the war was over.
He said: "You don't just walk to Berlin, and the sooner this country realises that the better."
The Normandy landings were the beginning of Operation Overlord - or the invasion of German-occupied Europe.
Originally planned to take place on 1 May 1944, the operation was postponed a month to allow time to gather more troops and equipment. The timing was important to allow for the right weather, a full moon, and tidal conditions.
To keep the destination of the landings secret, a deception plan Operation Fortitude was mounted which led the Germans to believe the main target was the Pas de Calais, much farther east.
When the landings finally began there were only 14 of the 58 German divisions in France facing the Allies. While there was stiff resistance at other beaches, Omaha was the only one where the success of the Allied mission was in serious doubt.
The invasion of Normandy was the largest amphibious assault ever launched. It involved five army divisions in the initial assault and over 7,000 ships. In addition there were 11,000 aircraft.
In total 75,215 British and Canadian troops and 57,500 US troops were landed by sea on D-Day. Another 23,400 were landed by air.
By 11 June the Allies had secured the Cotentin Peninsula beyond Cherbourg but progress continued slowly as the Germans put up fierce resistance. The end of the Normandy campaign came with the destruction of the German 7th Army in the Falaise pocket in August.
Although the Allies had reached the German frontier by September they decided to re-group during the winter, because of the failure of Market-Garden and the setback in the battle of the Bulge, and the invasion of Germany only began in January 1945.
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