The Battle of Greece is the continuation of the Greco-Italian War beginning with the German invasion of Greece to the fall of Kalamata in the Peloponnese. With the Battle of Crete and several naval actions, it is considered part of the wider Aegean Campaign.
Mare Nostrum & the roots of the Battle of Greece
Fascist doctrine had long emphasized the need for Italian colonial expansion and the reinstatement of Roman imperium over the whole of the Mediterranean, which the Romans had called Mare Nostrum - "our sea." Irridentist Italians wanted to recover the Italianate areas of Corsica, Savoy, and Nice from France, which looked like a strong possibility despite the Italian military's relatively meager contribution to the Battle of France. Italian possessions in Dalmatia and their conquests in Albania provided a good springboard for an attack on Greece, which Mussolini felt would be easy prey. The British were seen as preoccupied with protecting Libya and Egypt; along with East Africa, where the Italians had overrun British Somaliland and Abyssinia and were threatening Kenya and the Sudan.
The Greek Army, however, proved an able opponent, checking the Italian advance at several notable battles, including The Battle of Forty Martyrs, fought at a Greek Albanian village named for the Forty Martyrs of Sebaste. This neccesitated German intervention on Italy's behalf, something Hitler was willing to do, though it disrupted some of his timetables.
Germany strikes in the Balkans
On April 7th, 1941, the German Army invaded northern Greece, stunning the British and Greeks, and making them pull back their forces from deep into Albania to keep them from being flanked and destroyed. The Greek national sentiment was such that the army had to maintain a position along the "Metaxas Line", all along the northern border of Greece, near the port of Thessalonica. This was untenable from the start, and it was easy for the Germans to break through at multiple points with their Panzer groups. This, in turn, necessitated a Greco-British retreat further to the narrow pass at Thermopylae, where the Germans broke through again, all the way down until German forces were at the Acropolis. After some brief actions on the Peloponnese, the Greeks and British Commonwealth forces retreated to Crete. In the highly contested Battle of Crete, the Germans employed parachute forces and forced the British and New Zealanders off the southern half of the island, making Germany the dominant force in the Mediterranean.