  For each n>0, we define an algebra having many properties that one might
expect to hold for a Brauer algebra of type Bn. It is defined by means of a
presentation by generators and relations. We show that this algebra is a
subalgebra of the Brauer algebra of type Dn+1 and point out a cellular
structure in it. This work is a natural sequel to the introduction of Brauer
algebras of type Cn, which are subalgebras of classical Brauer algebras of type
A2n-1 and differ from the current ones for n>2.
