  The paper presents some recent results on the BMO Teichm\"uller space, its
subspaces and quotient spaces. We first consider the chord-arc curve subspace
and prove that every element of the BMO Teichm\"uller space is represented by
its finite composition. Moreover, we show that these BMO Teichm\"uller spaces
have affine foliated structures induced by the VMO Teichm\"uller space. By
which, their quotient spaces have natural complex structures modeled on the
quotient Banach space. Then, a complete translation-invariant metric is
introduced on the BMO Teichm\"uller space and is shown to be a continuous
Finsler metric in a special case.
