I'm not as experienced as a couple previous posters but have hiked in grizzly country somewhat. Have never had an encounter. Saw a grizzly once, all I could get was a pic of his rear as he ran away from me.
Generally, if you wear bear bells, refrain from carrying unsealed food with you, and carry bear spray - the risks are reduced very substantially that anything is going to happen. The one problem with bear spray is you cannot (legally) fly with it - not even in checked baggage. So you need to buy it on-site and then donate it when you leave. Don't buy it at your local REI and expect to take it with you to Yellowstone.
There are numerous stories of bear spray stopping a charging bear dead in its tracks - as previously mentioned, its important to hike in at least pairs with two people having spray. Spray is arguably as or more effective than a gun at deterring an attack in typical bear encounter scenarios, which typically involve close range where the bear is surprised by your presence and there is little time to react before the bear is close. (this scenario is entirely preventable with bear bells).
This is an awesome real bear encounter video:
https://www.youtube.com/watch?v=Iinv_5e_QGg
Also, make sure to pay attention to what the locals/rangers say about particular trails on particular days, etc. Generally, they know where the bears are and will close trails if necessary, etc.