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Like any large city, there is crime and certainly you have to be aware of you surroundings no matter where you are. Having that said, Zaragoza is pretty safe (compared to Madrid or Barcelona for example) and if you stay in the center you will be fine. Durning the day there should not be any need to worry as things are safe. At night if you stick to the well-lit areas and are with other people things will be fine. After hours, there are always tons of cabs running around and that is probably your best bet for getting around the city if you are not familiar with the city.
Zaragoza is like any other spanish city in that people stay out late, so on any given Saturday night for example (and especially in the summer months) you will see tons of people walking around the city center at 2,3 or 4 in the morning, going to bars, eating,etc.