Source: GADGET CITY Bill Jacobs
Suppose you hold: 
AQ1043 
4 
KQ32 
AQ2
You open 1 and partner responds 1NT. What now? Nothing fits. A jump to 3 is forcing, and whilst it could work out, it could equally get you into a mess at a high level. Conversely, a non-forcing 2 rebid is too little. Compromising with a non-forcing but invitational 2NT has problems of its own. Enter …
The Gazzilli Convention
Playing Gazzilli, you rebid 2. This shows one of two types of hand:
11-16, with spades and clubs,
or 17+, any distribution
After 2, responder bids 2 with any hand in the maximum range (about 8-9 points); with minimum values, responder makes a different natural bid – often a return to opener’s first suit.
Finally, after 1 – 1NT – 2 – 2, opener returns to 2 to show the spades+clubs minimum opener, or makes a different natural bid to show 17+ points (the partnership is now in a game-force).
A pleasant by-product of Gazzilli is that opener’s jump rebids show constructive values – about 15-16 points – and good distribution. For example, the auction 1 – 1NT – 3 depicts at least 5-5 shape, 15-16 points, and is non-forcing.
Gazzilli is an extremely popular convention in Europe – it is an Italian invention, as are so many gadgets in town – but it does come with some downsides. To start with, a final contract of 2 is no longer possible, nor can you end up in 2 after the 2 rebid. Nevertheless, most experts consider that the gains accrued with the improved definition of opener’s strong hands far outweigh the losses.
There are many different ways to play Gazzilli, and it can also be adapted to a system (such as 2/1) where 1NT is a forcing response. Some of these adaptations come under different names, for example Bart and Cole.
Gazzilli and its cousins should come in plain packaging with a Government health warning attached. It is not something that a casual partnership should use. There are many subtle aspects to the gadget and these require plenty of discussion and practice. I play the convention myself: it’s definitely not suitable for children.