Although Romans did slaughter thousands upon thousands of animals, both domestic and exotic imports, in the arena, they were also fond of exhibiting and displaying animals – many of which, sadly, were later killed in the arena.
Pliny the Elder, a Roman senator and author, is one of our major sources for exhibiting animals and tells us of the ill-fated attempt by the Roman general Pompey the Great to have harnessed elephants instead of horses in one of his three triumphs:
The first elephants seen in harness at Rome were those in the triumph of Pompey the Great over Africa, when they drew his chariot; this is said to have been done long ago at the triumph of Father Liber at his conquest of India. Procilius says that the elephants used at the triumph of Pompey were unable to go in harness through the gate of the city. In the exhibition of gladiators which was given by Germanicus, the elephants performed a sort of dance with rough, irregular movements. It was a common thing to see them throw arrows with such strength that even the wind could not make them change their course, to imitate among themselves gladiatorial fights, and dance the steps of the Pyrrhic dance. After this, too, they walked upon a tightrope and four of them carried a litter in which there was a fifth elephant, which represented a woman giving birth. Afterwards they took their place and so nicely did they manage their steps, that they did not so much as touch any of those who were drinking there.
It is a well-known fact that one of these elephants, who was slower than usual in learning his lessons and had thus been frequently beaten as a punishment, was found studying his lessons during the night. It is also very surprising thing that the elephant is able not only to walk up a tight-rope backwards but to come down it as well, with his head foremost. Mutianus, who was three times consul, informs us that one of these animals had been taught to trace the Greek alphabet and that he used to write in that language the following words: “I have myself written these words and have dedicated the Celtic spoils.” Mutianus states also, that he himself was witness to the fact, that when some elephants were being landed at Puteoli and were forced to disembark, terrified at the length of the platform, which extended from the vessel to the shore, they walked backwards to deceive themselves by forming a false estimate of the distance.
Pliny the Elder, Encyclopaedia 8.2-3
(For more on Romans and elephants see here.)
Lions were particularly popular in Rome from the first days they appeared there:
There was an ancient decree of the Senate, which prohibited animals being imported from Africa into Italy; but Gnaeus Aufidius, the tribune of the people, got a law repealing this passed and this allowed them to be brought over for the games of the Circus. Scaurus, in his ædileship, was the first who sent over parti-coloured panthers, one hundred and fifty in total; after which, Pompey the Great sent four hundred and ten, and the late Emperor Augustus four hundred and twenty. The same emperor was the first person who exhibited at Rome a tame tiger on the stage. This was in the consulship of Quintus Tubero and Fabius Maximus, at the dedication of the theatre of Marcellus, on the fourth day before the nones of May: the late Emperor Claudius exhibited four at one time.
The first Roman emperor, Augustus, made a habit of exhibiting exotic wild animals of all sorts to the Roman people.
Furthermore, if anything rare and worth seeing was ever brought to the city, it was his habit to make a special exhibit of it in any convenient place on days when no shows were being held: a rhinoceros in the Saepta [Julia], a tiger on the stage and a snake of fifty cubits in front of the Comitium and so forth.
Suetonius, Augustus 43.4
Links and bibliography:
George Jennison. 1937. Animals for Show and Pleasure in Ancient Rome
 We know it took place on March 12th, but are not sure if it occurred in 81,80, or 79 BCE.
 Bacchus, the god of wine, was said to have ridden in a chariot drawn by exotic animals as he spread his worship across the world when he made his first trip to Mount Olympus. He was often depicted on mosaics not just driving a chariot pulled by tigers and panthers, but surrounded by other exotic animals.
 Presumably the triumphal gate, which was only opened for triumphs. (We are not sure of its location.)
 A type of war dance, performed both in Rome and Greece.
 This was a very large snake: 50 cubits is roughly 23 metres in modern measurements.