Bellevalia - Bulbous perennial easily mistaken for the Grape Hyacinth
Beta - beet family. Best of which are the Swiss Chards - also grown as vegetables, but good ornamental plants as well.
Bulbocodium - Related to - and similar - the autumn flowering Colchicums. But this flowers in Spring.
Baccharis - Not worth chasing unless you want the herb version of Baccharis douglasii
Ballota - Interesting aromatic foliage - will not appeal to all - with silver and lime coloured versions.
Baptisia - The false Indigo, but still worth growing for its upright growth - fills a gap.
Barbarea - A cabbage-family herb that likes a damp position, in sun or dappled shade
Bellis - Best known for the small bedding daisies that flower in spring bedding schemes, and most owe their parentage to the common lawn daisy!
Bergenia - A very useful group of perennials, but often neglected and poorly grown. Simply because they are easy and forgiving. Large evergreen leaves, and pink variations in the flowers.
Bidens - These have become popular with the hanging basket types, but are useful - if short-lived - perennials for low growth.
Blechnum - One of many Hardy Ferns.
Bolax - Low growing and mat forming. Of use in rockery or small container.
Boltonia - Very similar to the Asters - Michaelmas Daisy group.
Borago - Borage - Bright blue flowered herb for dry places. Gives height. There is also a little-known dwarf type, that prefers a damp spot.
Bouteloua - Grass - Bouteloua gracilis lives up to its name, and is the only one worth considering for perennial useuage.
Bracteantha - Very attractive multi-coloured daisy flowers and a superb flower for the arrangers to dry.
Brassica - The ornamental Cabbages and Kale are rightfully popular, but not easy to fit in with perennial plantings!
Brimeura - Low growing bulbous perennial, often mistaken for a drooping bluebell.
Buglossoides - Low growing evergreen perennial with blue flowers.
Bupthalmum - Golden daisy flowers over low growing dark green evergreen foliage.
Bupleurum - Evergreen, low-growing, and some are classed as shrubs. But, a few good perennials if you can get them.