They had colonies else where, many of which eventually became part of the Ottoman Empire. The Genoese controlled not only, for example, the famous Constantinople neighborhood of Galata (just across the Golden Horn from the walled old city), but they also controlled much of Crimea, among other places. See the Wikipedia entry for Genoese colonies. You will still find Italians cropping in random places because of this. There are still ethnic Italians in Istanbul--my old landlord was one. His family had been in the city for over five hundred years. And Genoa wasn't the only pre-Unification state to develop colonies in the Mediterranean basin. Istria and Dalmatia were a well-known colony of Venice, though they eventually gotten taken over by Austria, and after WW2, the thousands of Italian Istrians were essentially ethnically cleansed from the region. But again, these were not the only Venetian colonies. I'm not aware of any other states that had colonies, but I would not be surprised if some others did at some time.

These were, however, primarily trade colonies rather than settlement colonies. Overseas settlement colonies seem to be something that largely comes about with the colonization of the New World. India and Egypt, for example, for the British were largely a trade colonies with relatively few British people settling there compared to the size of the local populations. The United States, Australia, and Jamaica operated as settlement colonies, on the other hand, with large population transfers (some of it obviously involuntary). Settlement colonies, to my knowledge, primarily occurred in areas where population density was relatively low which meant that the settlers had to compete for resources less/needed new settlers as labor to profitably work the land (there's an alternate theory, put forward by Daron Acemoglu, that the settlement colonies only occurred where the disease environment was favorable--mainly places without malaria). You'll notice that the places that did establish New World colonies--even tiny Courland--had direct access to the Atlantic (except for Russia, who obviously turned East and just kept going). Italy had no direct Atlantic access, and so set up colonies closer to home (Germany, Austria, the Ottomans, and Russia all primarily set up contiguous rather than overseas empires), and their "closer to home" was much more developed and densely populated than the New World.

I have always wondered and even posted on AskHistorians why all of Europe didn't colonize Africa instead of taking that long trip to the Americas? (no answer) I can understand that North Africa was largely desert and some of the tribes in the North were powerful but, being that it is the largest continent in the world why didn't they just go futher South?