Let D be the unit disc in the plane, with hyperbolic measure \(d\lambda\) and area measure dm. Let BMO be the space of functions on bounded mean oscillation which are the object of intensive study for the past twenty years. Let BMOH(D) stand for those f harmonic on D such that f is the Poisson integral of some functions in BMO and let BMOA(D) be the analytic functions in BMOH(D). In the obvious way, by integrating over balls, the author defines BMO(D,d\(\lambda)\) and BMO(D,dm) and their harmonic and analytic analogues for the disc. First, the author shows BMOA(D,dm) equals the Bloch space and a similar result holds for BMOH(D,dm). Then it is shown that \(BMOA(D)=BMOA(D,d\lambda).\) These concepts are then extended to the Riemann surface case, in the case R is of finite type he shows that a harmonic function with finite Dirichlet integral belongs to BMOH(R). However, there exists an infinitely connected plane domain R for which the inclusion fails. This contrasts with the BMOA(R) case in which it is known that every analytic function with finite Dirichlet integral belongs to BMOA(R) [see the reviewer, Canad. J. Math. 33, 1255-1260 (1981; Zbl 0468.30027)].