I live here in Fulham, right beside this beautiful place and have been visiting it for many years.
There are several aspects to it:
The park: the Palace sits in Bishops Park, which has just been comprehensively restored. The facilities are generally excellent, there are lots of tennis courts (look on the hammersmith and fulham website for the number to book one), a really good kids playground (with LOADS of kids, though) and lots of space to sit out.
Kids: if you have kids and if it's a sunny day, you're made. The area beside the Palace is fairly well enclosed with natural borders so it's easy to keep an eye on kids. On a sunny day you can join hundreds of locals and have a picnic on the grass. There's no charge to do this.
Cafe: Like most things London (and especially Fulham) the price is more remarkable than the quality of the food. There's an excellent terrace which is SouthEast facing and thus is sun drenched on nice days.
The cafe franchise holders turn over practically every year or two so maybe this latest one is going to crack it...
Museum: This used to be run by locals, and would not meet the usual expectations of what you might thing a museum ought to be. Visited it once and it felt a bit "home made" but maybe it has been upgraded too.
One other thing I learned yesterday, there's a church beside the Palace which was used in the Omen...it's actually really charming, maybe the demons have moved to Westminster?
It asks about fees below, there aren't any to visit the park and the grounds but if you go into the "museum" there is.
Makes a great day out.
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