[I read something very useful, so post here, hope it is useful for everyone who read it.]
Graphic showing correct depth to plant peony root stock First dig a hole. The depth of the hole depends on whether you're planting root stock or a potted peony.
Planting Root Stock
Peonies are most often sold as root stock (or peony divisions). These are literally pieces ("divisions") of peony roots. They usually have short pieces of cut-off stalk still attached. The depth at which you plant these pieces of root stock is very important. The growing buds (called "eyes" or "crown buds") should be no more than two inches (5 cm) below the surface of the soil. If peonies are planted too deep, they may not flower for several years (or, in some cases, may hardly flower at all)
Planting Potted Peonies
Sometimes peonies are sold in pots. These plants will generally already have some leaves showing — maybe even a few buds. Potted peony plants are usually sold in the spring, and it is fine to plant them as soon as you have purchased them.
Preparing the Soil
You could just dig a hole, pop in the peony, fill up the spaces with dirt, and hope for the best. Peonies are hardy and will survive all sorts of indifference and even abuse. However, if you want your peonies to thrive and look magnificent, you can improve their situation, as follows:
Graphic showing how to dig and fill in hole for planting peonies Dig a hole that is wider and deeper than you need for the plant alone. If the soil is poor, make the hole about 2 feet (60 cm) wide and 2 feet (60 cm) deep. At the very bottom of the hole put a layer of sand or gravel (unless your soil is naturally sandy or gritty). This provides drainage — peonies don't like their roots to be water-logged.
Then put down a thick layer (about 4"/10 cm) of compost, bone meal, or well-aged manure (or a mixture of all three!). This provides long-term "food" for the peony. (Some people will also add a handful of commercial fertilizer (10-6-4), but it's not really necessary. Also: fertilizer is very strong and can burn the peony if the fertilizer comes in direct contact with the roots of the plant.) Water this layer well, and wait until the water has drained away. This way you can tell for sure that the drainage is good. If the water stays in the hole for longer than 5 minutes, you should have put more sand in the drainage layer!
Mix the soil you dug up with some peat moss, compost, or other organic matter (Note: If your soil is naturally very heavy or "clayey", mix in sand as well as peat moss). Now add a little of this mix — enough to make a small mound, and put the peony on top of the mound; make sure to spread the roots out evenly around the mound.
Now put the peony in the hole. If you are planting root stock, measure from the "eyes" (i.e. growing buds) to the surface, and make sure the eyes are not too deep (i.e. not more than 2 inches (5 cm) from the surface). If they are, put more soil in the hole, and measure again. For potted peonies, the top of the soil in the pot should end up level with the surface of the soil in your garden.
When you are satisfied that the peony is at the right depth, fill up the hole with your soil mixture. Tamp down the soil, add more soil, if needed. Then, water well, and you're done!