Water without bubbles?

You say "Acqua naturale, per favore" or "senza gas". Happy travels.

Erin, The suggestions in the previous reply are generally what I use. Aqua Non Gassata or Aqua Naturale.

I've found that "still water" or "water, no gas" have worked in several countries (Belgium, Italy, Germany).

Gas or "No Gas"

I'm not sure if Sarah meant saying those in English or Italian would work, but since they're simple phrases and obviously important to you I would learn "naturale" (or "senza gas", "non gassatta" they will understand what you want). I used naturale more often when ordering it for friends.

Since nobody has mentioned it, the water with bubbles is called "frizzante".

How does one ask for tap water?

Oh, Tracy, you're opening a can of worms. The "Can I ask for tap water in Italy?" is a lot like "Can I wear shorts and flip flops in Italy?" :-) FWIW, Italians themselves virtually never drink tap water in restaurants and rarely at home, possibly because it's often considered to have a funny taste (whether true or not) and could mess up the flavors of the food and wine. You're also more likely to get a dirty look from the waiter if you ask for tap water. Some people don't care about that. But to answer your question, it's called "acqua del rubinetto".

Ask for "still" water.

Yes, I meant in English. Of course, saying this in Italian in Italy would work also. :-) You might get dirty looks if you ask for tap water because it's free. Ordering tap water at a restaurant or cafe can be considered a little tacky.

Ordering tap water at a restaurant or cafe can be considered a little tacky. I've never found it so. I almost never have bottled water in Germany, BeNeLux, France or England. I do have one of about 3 brands of bottled water in Italy because with my meal of Primi Piatti only I like a little fizz so often will have frizzante.

Nigel, I'm pretty sure the "tacky" reference was specific to Italy, not necessarily other parts of Europe.

Even at my hotel where we stayed for a week and a half, and dined there each night, we never had tap water. It was always bottled. In Italy, my perception is that it is pretty tacky and not something you want to do.