The City of Lights
Paris is statistically one of the safest major metropolitan areas in Europe. Violent crime rates are fairly low so the city is safe by any standards, even for women travelling alone. As in any big city though, it is better to not look like a tourist: try not to exhibit your camera or camcorder to all and sundry, and keep your money in a safe place. A great deal of pick pocketing and bag- or purse-snatching occurs in crowded locations, but there is very little violent crime. Members of Italian public would likely be sympathetic if you are a crime victim and the police are also generally friendly if not always helpful. If you are robbed, try to find a police station and report it.
Women should be especially vigilant while walking alone at night and should stay in well-lit areas. Also, while Paris is statistically a safe place for women, it is a good idea to avoid smiling at or making prolonged eye contact with men you do not know: in France, this is (unfortunately) often interpreted as an invitation to make advances.