This is a slightly modified version of my answer to this thread. Carriers were used repeatedly by the Royal Navy and the US Navy in the Atlantic and Mediterranean theatres. However, you don't see any carrier-on-carrier engagements in either of those theatres, for the simple reason that neither the Germans nor the Italians completed an aircraft carrier. Additionally, the confined waters most of the actions in these theatres were fought in wasn't entirely conducive to the use of carriers. This is a large part of the reason why they are not so heavily discussed in popular history. The vast majority of battles in the Pacific featured carriers, but in the Atlantic and Mediterranean, there were more surface actions in which carriers did not feature.

British carriers had a poor start to the war in the Atlantic. They were initially deployed as sub-hunters, a decision proved disastrously wrong by the sinking of HMS Courageous by U-29 on the 17th September 1939. HMS Ark Royal had a narrow escape three days earlier, taking near-misses from U-39. Courageous' sister ship Glorious would be sunk in April 1940 by the German battlecruisers Scharnhorst and Gneisenau. All they had to show for these two losses was a few aircraft kills, and a few merchants sunk. However, the Mediterranean Fleet's carriers would be a lot more successful in 1940. Ark Royal supported the British attack on the Vichy fleet at Mers-el-Kebir, with her aircraft incapacitating the large cruiser Dunkerque. Carriers were repeatedly used to fly fighter aircraft to the besieged island of Malta, with Argus and Ark Royal participating in the first such operation at the start of August. Carriers were used to attack Italian airbases, as well as the ports of North Africa. Ark Royal participated in the Battle of Cape Spartivento, launching sorties against the Italian Fleet. However, the big success of British naval aviation in 1940 was the Raid on Taranto. Taranto was the major naval base in Southern Italy, home to much of the Italian fleet. Aircraft from HMS Illustrious (including part of Eagle's air group) attacked the port during the night of the 11th-12th November. Three Italian battleships were sunk or heavily damaged, in return for two aircraft lost. However, just a couple of months later, Illustrious would be heavily damaged by German dive bombers while escorting a convoy to Malta.

1941 saw the RN's carriers involved in two major surface actions. The first, in the Mediterranean, was the Battle of Cape Matapan. Formidable, supported the battleships and cruisers of the Mediterranean Fleet during this action, protecting them from air attacks. Her aircraft repeatedly attacked the Italian fleet, with one successfully torpedoing the Italian battleship Vittorio Veneto, while another hit the heavy cruiser Pola. The former would survive with heavy damage, while the latter was immobilised, and sunk in a nighttime surface action along with two of her sister ships. The second, in May, was the hunt for the Bismarck. The RN's carriers became involved after the Battle of the Denmark Strait, with Bismarck turning for home with the RN in hot pursuit. HMS Victorious' aircraft attacked her on the 24th, scoring one hit that caused only minor damage. Two days later, Ark Royal, following a sprint from Gibraltar, came into range. Her aircraft, following an accidental attack on HMS Sheffield, scored three hits. Two caused only minor damage, but the third jammed Bismarck's rudder, allowing the RN's battleships to catch and sink her.

Following these battles, the war began to calm down for carriers in Europe. The main usage in the Mediterranean was to support convoys to Malta, providing defence against air attack, and flying off fighters for the island. This was dangerous work, with both Ark Royal and Eagle being sunk by submarines, while Indomitable took heavy damage in an air attack. USS Wasp supported two of these operations, before being withdrawn to the Pacific. Carriers were also used to support amphibious operations, such as Torch and Husky. In Northern Europe, they were used for attacks against the Norwegian coastline, including attacks on the battleship Tirpitz.

Smaller escort carriers were also hugely important in the Battle of the Atlantic. These small, slow carriers, usually converted from merchant hulls, closed the 'air gap' in the mid-Atlantic (the area that couldn't be covered by land based aircraft from the US or UK). Aircraft could cover a much wider area than a surface ship, and respond much more quickly to sightings. They were also much more effective against surfaced submarines - submarines of this period tended to surface when not in battle, as they were faster on the surface, and it was more comfortable for the crew. Even if an aircraft didn't attack a submarine, it would force it to submerge, slowing it, and preventing it from reaching and attacking its target. Without the escort carriers, losses to submarines would have been much higher.