Is it safe? Generally, statistically—yes.
Is it safe for a woman traveling alone? Generally, statistically—yes—with normal precautions.
Is it safe for your daughter? You would know better than any of us. Does she tend to be over-trusting or naive? Would you feel comfortable having her travel to NYC alone? Does she have some street-smarts, keep her wits and not get flustered? This would go for anyone—male or female.

…and, how old is she? Statistically, most cities in Western Europe are safer than large US cities. Trains are very safe as well. If you have no problem letting her be by herself for a week in a city in the US, then she should have no problem in Europe.

I ditto Don and Mim about general safety. She’s probably safer in Europe than on most US college campuses. My concern: Is she comfortable with the idea of traveling alone and does she have some experience getting around big cites? If she’s worried about safety, uncomfortable navigating a new city, likely to over worry about getting lost, worried about making travel arrangements; it won’t be a great experience until she gains some level of confidence as an urbanite and as a traveler. How much has she traveled? Has she navigated public transit in a big city? Checked into a hotel? Dealt with airport security, flight delays and the general hassles of a long flight or complicated travel connections? My rural/suburban son could have graduated from high school without ever crossing a street if we hadn’t traveled.

While personal safety shouldn’t be an issue, there are some concerns about safety of your stuff. Pickpockets and purse-snatchers target tourists, so you should review safety procedures: carry only a small amount of cash in pocket or purse, keep larger amounts of cash, credit and debit cards in a money belt or at least split things up so it wouldn’t be easy to get everything. Look confident, duck into a store or cafe if you need to look at a map. Don’t flash wads of cash or expensive electronics, lock up stuff at your hostel. Pay attention to your stuff on trains and in train stations.

Special advice for solo women. — problems are rare, but better avoided in the first place. A friendly smile and hello to a stranger is the norm on many college campuses. It’s not the norm everywhere in Europe and may be interpreted as a come-on. Dark glasses can help avoid eye contact that may be misinterpreted. If someone pays too much attention firmly and politely tell them to leave you alone. If the problem persists raise enough fuss to attract attention.

Most important bit of advice for both personal safety and safety of possessions: STAY SOBER.

I sincerely and strongly advise of having a companion at all times – makes it fun, safe and two minds better than one.
Stay in package tours.
Stay at City centers. Not to flaunt too much cash or behavior. Avoid late hours at bars or restaurants unless it is next to your hotel etc.
Register at US consulate in all countries. Call for an 1-1 consultation with US citizen services and get all brochures etc etc. Be mindful of the fact that offers and guides at the US consulate are registered services but expensive. So take advise from forums such as Euro trip or fellow classmates to avoid extra expenses.
Be mindful of the fact that unnecessary touts frequent around the US consulate or US libraries.
In Central London, it is very safe and sound.

Sam – totally disagree with you on many of these points. No reason to stay in package tours. Most European cities are safer than US cities. The best word of advice is not to walk back to your hostel loaded, by yourself at 4 in the morning drunk.

Agreed!!!
I don’t have a daughter so I may not be the best advisor out there.

My niece just returned from Thailand with her best friend. She is 15 and went around everywhere. However, she got bored as her friend didn’t want to go out anywhere after dark. Now that’s what I call a responsible person.
My niece didn’t like it simply because her best friend spent the entire evenings surfing the net. That’s what i call an useless companion.