This paper concerns certain generalizations of BMO, the space of functions of bounded mean oscillation. Let \(\rho\) be a positive nondecreasing function on (0,\(\infty)\) with \(\rho (0+)=0\). A locally integrable function on \({\mathbb{R}}^ m\) is said to belong to BMO(\(\rho)\) if its mean oscillation over any cube Q is O(\(\rho\) (\(\ell (Q)))\), where \(\ell (Q)\) is the edge length of Q. Carleson measures are known to be closely related to BMO. Generalizations of these measures are shown to be similarly related to the spaces BMO(\(\rho)\). For a cube Q in \({\mathbb{R}}^ m\), \(| Q|\) denotes its volume and R(Q) is the set \(\{(x,y)\in {\mathbb{R}}_+^{m+1}:\) \(x\in Q\), \(0<\ell (Q)\}\). A measure \(\mu\) on \({\mathbb{R}}_+^{m+1}\) is called a \(\rho\)-Carleson measure if \(| \mu | (R(Q))=O(\rho (\ell (Q))| Q|)\), for all cubes Q. L. Carleson proved that a compactly supported function in BMO can be represented as the sum of a bounded function and the balayage, or sweep, of some Carleson measure. A generalization of this theorem involving BMO(\(\rho)\) and \(\rho\)-Carleson measures is proved for a broad class of growth functions, and this is used to represent BMO(\(\rho)\) as a dual space. The proof of the theorem is based on a proof of J. Garnett and P. Jones of Carleson's theorem. Another characterization of BMO(\(\rho)\) using \(\rho\)-Carleson measures is a corollary. This result generalizes a characterization of BMO due to C. Fefferman. Finally, an atomic decomposition of the predual of BMO(\(\rho)\) is given.