Unfortunately, yes, they do - the flowering stems need staking because, as you've noticed, the flowers are too heavy to be self supporting.

I've never staked before, any tips, group all the stems together with on stake, or into groups?

Individually I'm afraid - its probably all the rain this year, lots of lush, taller than usual, soft and sappy growth on everything...

Yep - lots of green in the garden, but I like this time of year just as the garden is coming into colour!! Thanks for the advice.

I use supports that consist of a metal grid with legs (see www.babyllon.co.uk/product-catalogue/plant_supports - which they call 'Grow Through Supports').
I'm not advertising this supplier, it's just I found a photo of what I use from doing a Google search

Some of the species peonies do not need staking we have P. veitchii which even in the wettest of summers does not need staking as the flower heads are small and single.

I support all mine with those grow through supports similar to what Andrew is describing. They need put into place in spring so the plant can grow up through.
If I can offer a wee tip....I position mine quite low to the ground then as the plant grows taller I raise the height a wee bit. Thus I find it keeps the plants stems closer together which in turn offers a bit more support

I use similar metal grid supports on mine, Marcus, and this year, even those have been a bit inadequate. I tried looking online for more, bigger, robust ones but have been scared off by the very high prices, so next year it will be four or five stout bamboo canes driven into the soil, laced around with garden twine, and added to as the paeonies grow bigger. Not quite so aesthetically pleasing, but once the foliage has got going, I doubt anything much of the canes will be visible. If I had staked all the blooms on mine individually, I should have run out of canes very quickly, and personally, I don't find it's necessary.