In a hard rain storm, I get water dripping in my fireplace. I've never noticed this in other houses. What keeps rain from coming down a chimney? I've seen bird in a chimney on occasion--if they can come in, why couldn't rain? And what might be wrong with mine now?

Get a chimney cap.

Chimney caps fit the tops either attached to the flue liner or the outer perimeter of the brick,depending on if you have a flue exposed enough to attach one too.

Many styles,many colors/finishes,prices.

Easy to install,as long as the chimney is in an accessible spot(say in the center of the roof,and the roof is walkable),if not,a roofer/handyman can easily install one.

This link may be helpful. All About Chimney Caps
Good luck, Tom

More than likely a chimney cap, however you may want to look at the mortar and brick around the top half of your fireplace just to make sure you don't have major deterioration and the leak inside the fireplace is not a secondary symptom to a major problem.

Deuces wild brought up a real issue as to a possible cause. The Mortar wash is most likely bad. If this is a conventional fireplace the inside of the chimney is offset from the Damper area and it takes a lot to get that wet.

I an old schooler and build fireplaces and I'm Against the Metal caps they sell.

Thoses Caps will cut the air flow down and can effect the damper fireplace damper settings The best way to cap a Fireplace chimney is lay 3 or 4 courses of brick on each chimney corner and mortar a flagstone cap on.

This will actually IMPROVE your air flow in a chimney. When there is a cross wind it hits the lower sloped masonry wash and deflects the wind upwards towards the bottom of the flag cap and then sends it out.

Look at it this way. When power boating and there's water in the back of the boat while moving the boat along you can pull the boat plug that's under the water line and it sucks out the water that's in the boat.

This is what a flag cap will do for a chimney wind "pulling" the air out of the flue..