Cassius Dio tells us that when the Romans invaded Britain that Claudius used war elephants in the campaign.

Shortly after Togodumnus perished, but the Britons so far from yielding stood together all the more closely to avenge his death. Because of this fact and his previous mishap Plautius became alarmed, and instead of advancing farther proceeded to guard what he had already gained and sent for Claudius. He had been notified to do this in case he met with any particularly stubborn resistance, and a large reinforcement for the army, consisting partly of elephants, had been assembled in advance.

Other than this direct source from Cassius Dio I don't have any other quotes from historians but if I remember correctly most are pretty sceptical of this claim for obvious reasons - transporting elephants to Britain, their actual effectiveness and Claudius being able to orchestrate a battle are just a few of the concerns. If you're ever in the east of England it might be worth digging for some elephant bones though!

Depends on what you mean by "the Romans". Indeed, the Roman army used elephants sporadically. However, there were no ethnic Roman elephant handlers. The Romans simply hired the already trained Numidian and Mauritanian elephant handlers from North Africa, like the Carthaginians did before them. Numidia and Mauritania were Berber monarchies vassals of Rome before they became Roman provinces in 47 BC and 40 AD, respectively.

The Romans deployed Numidian elephants once during the Celtiberian Wars in Spain (181-133 BC). It was a colossal failure, as described in Appian's The History of Rome:

The Arevaci convened immediately, even in the night, at Numantia, which was a very strong city, and chose Ambo and Leuco as their generals. Three days later [the praetor Quintus Fulvius] Nobilior advanced and pitched his camp at some four kilometers from the place.

Here he was joined by 300 horse and ten elephants sent to him by Massinissa [King of Numidia]. When he moved against the enemy he placed these animals in the rear where they could not be seen. Then when battle was joined the army divided and brought the elephants into view. The Celtiberians and their horses, who had never seen elephants before, were thunderstruck and fled to the city.

Nobilior advanced at once against the city walls, where the battle raged fiercely, until one of the elephants was struck on the head with a large falling stone, when he became savage, uttered a loud cry, turned upon his friends, and began to destroy everything that came in his way, making no distinction between friend and foe. The other elephants, excited by his cries, all began to do the same, trampling the Romans under foot, scattering and hurling them this way and that. (This is always the way with elephants when they are enraged. Then they take everybody for foes; wherefore some people call them the common enemy, on account of their fickleness.)

The Romans took to disorderly flight. When the Numantines perceived this they sallied out and pursued them, killing about 4,000 men and three elephants. They also captured many arms and standards. The loss of the Celtiberians was about 2,000.

There was another question on war elephants a few months ago and I mentioned that Cassius Dio tells of the presence of one elephant in Caesar's army during his invasion of Britain (55-54 BC). However, other contributors replied that this was probably a confusion on Dio's part, and that the only time Romans brought elephants to Britain was during Claudius' campaign in 43 AD, referenced by AberStans.

Actually yes, Rome effectively deployed some 20 elephants against the Macedonians at the battles of Pydna and Cynoscephalae. They got them from Numidia.

Source: War Elephants, Kister 2006.