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Peony or paeony is a name for plants in the genus Paeonia, the only
genus in the flowering plant family Paeoniaceae. They are native to
Asia, southern Europe and western North America. Boundaries between
species are not clear and estimates of the number of species range
from 25  to 40.
Most are herbaceous perennial plants 1.5 - 5 feet (0.5 - 1.5 metres) tall, but some resemble trees up to 5 - 10 feet (1.5 – 3 metres) tall. They have compound, deeply lobed leaves, and large, often fragrant flowers, ranging from red to white or yellow, in late spring and early summer.
The peony is named after Paeon (also spelled Paean), a student of
Asclepius, the Greek god of medicine and healing. Asclepius became
jealous of his pupil; Zeus saved Paeon from the wrath of Asclepius by
turning him into the peony flower.
The family name "Paeoniaceae" was first used by Friedrich K.L. Rudolphi in 1830, following a suggestion by Friedrich Gottlieb Bartling that same year. The family had been given other names a few years earlier. The composition of the family has varied, but it has always consisted of Paeonia and one or more genera that are now placed in Ranunculales. It has been widely believed that Paeonia is closest to Glaucidium, and this idea has been followed in some recent works. Molecular phylogenetic studies, however, have demonstrated conclusively that Glaucidium belongs in Ranunculaceae, but that Paeonia belongs in the unrelated order Saxifragales.
A cultivated hybrid peony
Peonies can be classified by both plant growth habit and flower type. Plant growth types are Herbaceous (Bush), Tree, and Intersectional (Itoh).
Each flower type becomes more complex in its arrangement of petals. The flower types include Single (e.g., Athena, Dad, Krinkled White, Scarlet O’Hara, Sea Shell), Japanese (Nippon Beauty, Madame Butterfly), Anemone, Semi-Double (Paula Fay, Coral Charm, Miss America, Buckeye Belle), Double (Ann Cousins, Gardenia, Kansas, Paul M. Wild, Tourangelle) and Bomb-Double (Red Charm, Raspberry Sundae, Mons Jules Elie). Herbaceous peonies die back in winter and regrow in spring, while tree peonies lose their leaves in winter, but leave woody stems.
Intersectional (Itoh) peonies are hybrid crosses between tree and herbaceous types. They have the large flowers of a tree peony, but die back to the ground each year like herbaceous peonies. 
Chemistry and biological activities
Over 262 compounds have been obtained so far from the plants of
Paeoniaceae. These include monoterpenoid glucosides, flavonoids,
tannins, stilbenoids, triterpenoids and steroids, paeonols, and
Biological activities include antioxidant, antitumor, antipathogenic, immune-system-modulation activities, cardiovascular-system-protective activities and central-nervous-system activities.
The herb known as Paeonia (Bai Shao, Radix Paeoniae Lactiflorae), in particular the root of Paeonia lactiflora has been used frequently in traditional medicines of Korea, China and Japan. Research suggests that constituents in Paeonia lactiflora - paeoniflorin and paeonol - can modulate IgE-induced scratching behaviors and mast cell degranulation. 
Herbaceous and Itoh peonies are propagated by root division, and sometimes by seed. Tree peonies can be propagated by grafting, division, seed, and from cuttings, although root grafting is most common commercially. 
Courtesy of Wikipedia