Eucalyptus badjensis - Badja Gum, Big Badja Gum, Gully Ash (New South Wales) This is a large, narrow-leafed species with bark that is rough and fibrous at the base, but smooth and light colored on the higher parts of the trunk and branches.  It grows at reasonably high elevations and can be expected to tolerate fairly severe frosts.  In addition, it is also an extremely fast-growing species.  It prefers deep, moist soils to grow best.  The juvenile leaves look like those of E. viminalis.  Young plants have withstood 11°F in New Zealand, but older trees were heavily damaged from 7°F and did not recover well.  8 to 14°F.
Photos of Eucalyptus badjensis

Eucalyptus baeuerlenii - Baeuerlen's Gum (New South Wales) A rare, hardy species with very dense foliage.  It often has a mallee habit but it is likely to grow into a medium sized tree in cultivation.  The bark is very interesting and can range in color from red to grey-red to cream-green to brown.  Coming from high mountain slopes, it has proven completely hardy to 7°F and is therefore worth growing throughout zone 8 and should be tested in colder areas.  4 to 12 (?) °F.
Photos of Eucalyptus baeuerlenii

Eucalyptus barberi - Barber's Gum (eastern Tasmania) A small tree or mallee closely allied to such larger swamp gums as E. brookeriana and E. ovata.  As it grows a ways inland in the rain shadow of western Tasmania's mountains, it may be somewhat tolerant of summer frosts and some heat, but it also withstands poor drainage.  Unlike E. brookeriana, the bark tends to be smooth and white, except occasionally near the base where rough grey bark may persist a few feet up the trunk.  Very young trees quickly develop widely spaced, weeping green leaves, and have a moderate growth rate.  Should be tried in zone 8.  10 to 16°F.
Photos of Eucalyptus barberi

Eucalyptus bensonii - Mountain Swamp Gum (Australia) Probably at least as hardy as related E. ovata.

Eucalyptus benthamii - Camden White Gum, Dorrigo White Gum, Nepean River Gum Extremely fast growing, attractive, and quite hardy tree, yet little known in cultivation.  Probably hardy to around 5 to 13°F.

Eucalyptus blakelyi - Blakely's Red Gum, Blakely's Gum, Forest Red Gum  Despite optomistic initial reports from England, I do not think this tree is well suited to maritime climates.  It is related to E. camaldulensis and probably no hardier.  Possibly useful in Texas and the Southwest as is E. microtheca.  14 to 21°F; probably lower in a dry climate.

Eucalyptus blaxlandii - Blaxland Stringybark  A medium stringybark that tolerates dry conditions.  Probably hardy to around 6 to 14°F.
Photos of Eucalyptus blaxlandii

Eucalyptus bridgesiana - Apple Box, Moonbi Apple Box, Apple, Apple Gum, But-but, Swamp Apple(eastern Victoria, eastern New South Wales, and southern Queensland) Simply put, the juvenile foliage of this species is just great: the alternate leaves are perfectly heart-shaped, closely packed along the stem, and on many plants an irresistible shade of blue, unsurpassable as a cut foliage plant.  It also makes a nice tree where hardy, though the main branches often tend to grow rather stiff and crooked.  It is very well adapted to poor clay soils and grows rapidly to a height of 60'.  It eventually develops a large open crown of pointed leaves held in dense tufts.  9 to 15°F.
Photos of Eucalyptus bridgesiana

Eucalyptus brookeriana - Brooker's Gum, Rocka Rivulet Gum (Victoria and Tasmania) I would be curious to know the origins of the common name of this species.  It makes a large tree with bright green leaves.  The outstanding feature is the bark, which is usually fibrous near the base, but above 15' it is shed annually to reveal shades of orange-red, green, bronze and cream.  It is one of the fastest-growing species in perpetually cool, wet climates and should be grown on the west coasts of the British Isles.  It also tolerates rather poor drainage. Although it does not grow natively in exceptionally cold areas, it is said to be quite adaptable and can probably withstand temperatures down to 10°F.
Photos of Eucalyptus brookeriana

Eucalyptus burgessiana - Faulconbridge Mallee Ash (Blue Mountains, New South Wales) This very rare large mallee is currently under threat from urban expansion.  It has thick, dark green leaves and smooth bark that is mostly grey but can have tan or orange patches, and even patches which contain small spots!  Related to E. obtusiflora, E. apiculata and some other hardy species; guessing hardy to somewhere around 10°F.