Florence: Any parks/outdoor space for kids to run around?

I think that Piazzale Michelangelo might have what you're looking for. And, it has spectacular views of the city. It's south of the Arno River.

I haven't been there, but we're staying at Hotel David, right near there, this July. So, others that are more familiar with the area can chime in.

Boboli Gardens are beautiful and will give the kids some space. Whilst it is not a vaste expanse of grass, there are enough areas for the children to run around, up and down. Great views from the 'top'. Explore Altrano. Perhaps even use GoogleEartth to look for green spaces? Take the train to Lucca for the day. The walk around the ramparts is lovely, and there are bike riding options.

Kid play in the piazzas all the time. Chasing pigeons is a lot of fun.

Rick's Florence & Tuscany book has a chapter "Florence With Children", here are just a few of his tips: If you're there in the summer, avoid the midday heat. Public bathrooms are hard to find, don't pass one up. The open air markets give kids a chance to explore a bit though they aren't parks. Piazza d'Azeglio. Kids Museum. The Piazzale Michelangelo (suggested in another post) is too far to walk from your hotel with the kids, it's uphill after the river.

Up to Pza Michelangelo is not too far for kids at all unless they are toddlers. Kids have boundless energy. And they will sleep well. Have some water and fruit for a snack when you get there. The church of San Miniato almost next door. Beautiful, and if you are there about 5:00pm, you can hear the monks chanting. The view from here is as good as Pza M. and far less crowded. The monks have a little shop next to their church. Don't forget to look at the Torre San Niccola (if I remember correctly)on the way back. AA (British) have great books on Italy. Walks, car tours, regions and often include a little 'what the kids might like' section. Check a good book store.

Fort Belvedere is next to the Boboli Gardens. Kids will like the fort, walls, and the views. Plenty of room to walk/run around and the adults will really enjoy the views. It's actually accessible across from the Boboli Gardens but we wound up walking up a big hill to get there. Well worth for the view.

Thanks everyone for the great suggestions!