Nara Park is one of my favorite places in Japan. It's inhabited by deer that are so used to the humans that they only reason they run away from a person is because they're annoyed rather than afraid. In fact, it's more common to see people running away, after attempts to feed the deer with "shika sembei" (deer crackers) attract hordes of their brethren looking for a free nibble. The Park is also home to a few structures of historical importance.

Although not strictly part of the Park, Todaiji, where you can see Japan's largest Buddha statue, is only a short (maybe 20-30 minutes) walk away. There are plenty of deer to play with along the way.

Park full of historical sites, museums and off course deer. You can buy deer cookies for 150 yen. Be careful which deer you give it to, some of them can be very annoying and nearly all of them are pushy. But it is great fun to watch the deer chase the tourists. Not recommended to hold a picknick in this park, our four legged friends will ruin it for sure.

Fantastic side trip from Kyoto (1 hour away) where you can see several World Heritage Sites in one park. Best way to see it is to rent a bike next to the train station for $5 a day (frickin' bargain). Lots of deer also roam the park, which is fun until they chase and bite you.

You have never seen this many deer in one place in your whole life, i guarantee it. And they're all so tame, they'll just walk right up to you for biscuits XD

The Nara city is world heritage.
There are many quite beautiful temples and shrine.
My recomend is KASUGATAISYA.