The British expected that the tank would dominate the Crusader Battle (also known as the Winter Battle). While the Official History indicates that the Germans also thought that, Rommel certainly did not. Rommel was a combined armed advocate, and made his reputation with infantry, using infiltration tactics pioneered in WWI. His approach to modern combined arms battlefighting was to use "sword and shield" tactics. He used armour against infantry and transport and artillery against tanks, particularly the 5cm PAK38 and the 88mm FLAK18 and other models.
The Germans had their pre-war tank designs. There were a small number of Pzkw I's, which were just MG armed and were of little consequence. The larger 20mm gun-armed Pzkw II was useful as a reconnaissance tank. The Pzkw III was the main battle tank, although the newest were just armed with a medium length 5cm gun that was capable of firing the "arrowhead" ammunition. The Pzkw IV was used in a similar role to the British support tanks, in that it was burdened with the short 75mm gun, intended to fire HE shells. The German tanks had the advantage of having some face-hardened armour that would shatter the British 2pdr shot and had a few Pzkw III's with appliqué armour.
The British had tanks built to an erroneous, pre-war concept. They had the slow, well-armoured infantry tanks. Originally, these were the Inf. Mk.II Matilda. Later, they were joined by the Inf. Mk.III Valentine, which were capable of being upgunned, unlike the Matilda, stuck with its cast hull. Infantry tanks were supposed to support infantry, but lacked a suitable armament for that role. The faster cruiser tanks were to fight other tanks, but they were under-gunned and armoured to be successful in that role. It was only when the British saw the American Grant, Lee, and Sherman, that they saw how to effectively arm a tank. The American medium tanks had the medium velocity 75mm gun at this date, which could fire either HE or an AP shot. The AP shot was on the order of 12 lbs, so it had much greater striking power than the high velocity, but lightweight 40mm 2pdr gun. Due to bad decision-making, the excellent 57mm 6pdr gun did not appear until 1942.
The Italians had the diesel-powered M13/40, at first. This was joined by the M14/41 and M15/42 tanks. The latter with a longer, higher velocity 47mm gun. They also had the useless L3 light tanks. Their more useful light tank was the L6, which really had too high a silhouette. They did have the excellent Autoblinda armoured cars, which were as good as any used by the Germans and better than what the British had.