Young Billy Webb likes to swap things with his schoolmates. He meets Alfonso Bonzo, an 'Italian exchange student' who also has a talent for swapping things (thus his self-description: he is an Italian student who exchanges things). Billy makes a series of magical but increasingly risky swaps with Alfonso, leading to unexpected and sometimes sinister consequences as Alfonso's true nature becomes questionable.
While in hospital, reporter Trevor Trotman meets Billy Webb, a young boy with a broken leg. Billy regales Trevor with how he came to meet the mysterious Alfonso Bonzo. Billy is the swap king of his school, always able to make a deal with someone or other. Then one day, a mysterious Italian exchange student turns up on his doorstep with an offer of a swap of his own.
Billy finds himself in a dilemma when he's due to hand in a project to his teacher Mrs Peasgood, which is in the bag he's swapped with Alfonso. But while he is easier to find than expected, Alfonso isn't as forthcoming when it comes to swapbacks.
Concerned by the power of the mysterious bag, Billy returns it to Alfonso, who has a new offer: swap his dog Fred in return for a Italian greyhound called Giulietta for the day. What's more, it seems that Giulietta is no ordinary greyhound; she can do tricks and soon Billy is showing her off at school. However, it also attracts the attentions of school bully Ginger Gahagan and risks discovery by headmaster Mr Hardwood, when a school assembly goes horribly wrong.
Mr Hardwood has confiscated Giulietta, leading Billy to wonder just how he can get her back. But just when things start to go right for him, Giulietta runs off, leading Billy with the uneasy feeling that Alfonso – still in possession of Fred – is behind it all. But things get stranger when Fred turns up — and can talk!
Billy tries to tell his sceptical family about Alfonso, at least until Alfonso unexpectedly turns up and proceeds to ingratiate himself into the family, while turning his attention to Billy's older sister Linda. When Alfonso uses his powers to meddle with the TV, he offers to bring over his own set as a replacement. But his Milanese TV set has some very peculiar side effects, which Billy is soon to discover.
Billy finds himself on telly, playing Thunderbolt Thorpe in the World Snooker Championships. His concerns intensify when Linda returns early from the disco and says that Alfonso wished to be her brother. Billy has forgotten to turn off the TV, which puts himself and his family in peril.