the genus allium belongs to the allioideae subfamily of the amaryllidaceae (amaryllis) family. some of its members like a. schoenoprasum (chives) or a. ampeloprasum (broadleaf) are popular food plants. others are being used as ornamentals.
many of the allium species originate from asia and europa. but the genus contains also members native to central/south america and south africa. a list of “all” species you can find here.
not all alliums are forming bulbs. some species are growing from a rhizome (underground stolon) or tuberous roots.

the flowers of allium ursinum (ramsons).

allium schoenoprasum (chives)

allium cristophii is commonly known as persian onion or star-of-persia.

allium oreophilum

allium tuberosum (chinese chives)