A fanny pack is safer than a shoulder bag, but it's still an attractive and easy target for pick pockets in crowds, like in museum lines and on buses, and most of them can be cut through. I buy travel clothes with lots of exterior and interior pockets -- I like the ones made Tilley, which have the added advantage that many of the external pockets are velcro-sealed, meaning they make a noticeable noise if someone opens them.
I also sometimes use pouches that have a big loop, big as the pouch itself, you can put your belt through. These of course can be cut through, but they probably won't be cut off, since the large size of the loop means it would actually be harder to cut it off your belt than to cut through it. You can wear these in front, where you can casually keep your hand over it when you're in a crowded place.
If you really need to carry something in a bag, and you're not in a place you know is safe, it's best to carry your stuff in a rather used-looking plastic or even paper shopping bag from a medium-priced (not luxury) local store. This suggests that you're a local, and that what you have in the bag is more likely to be a salami sandwich and a bottle of mineral water than a passport and camera.