Training and intelligence
They are intelligent, but certain aspects of training may be dfficult. Housetraining can proove hard. As with other sighthounds, Italian Greyhounds are easily distracted by movement, so a reliable recall can be tough to achieve.
Italian Greyhounds are very cuddly by nature - this is partly due to affection, and partly a need for warmth. They can take time to bond with you, but once they have, it will be a close bond.
Attitude towards strange dogs and people
Italian Greyhounds can be shy with other people and dogs. Socialisation is vital to avoid problems.
Grooming and shedding
The Italian Greyhound's short, light coat sheds little, and grooming will be minimal. The thin coat does leave them vulnerable in cold weather.
Exercise should be restricted up to 18 months of age. Italian Greyhounds have bursts of energy, and needs an outlet to burn it off, but then will be happy to relax at home.
Need for company
They need company, and don't like to be left alone. If you can't provide the company all the time yourself, consider getting another Italian Gray hound - they do usually prefer to live in pairs - or even a cat. (Opinions vary on cats and Italian Greyhounds, but many live perfectly happily together.)
Daily teeth cleaning is important for Italian Greyhounds. For some reason these dogs do well in pairs; they like the company of their own breed. They are very delicate dogs, and that gorgeous velvety coat just has to be touched to be believed.