The author completely characterizes the boundedness and compactness of the weighted composition operator \(W_{\psi,\varphi}f=\psi (f\circ \varphi)\) acting on \(BMOA\) and \(VMOA\) of the unit disc. The results extend and unify those known for the cases \(\varphi(z)=z\) and \(\psi(z)=1\) corresponding to the multiplication operator \(M_\psi\) [see \textit{S.\,Janson}, Ark.\ Mat.\ 14, 189--196 (1976; Zbl 0341.43005) and \textit{D.\,A.\thinspace Stegenga}, Am.\ J.\ Math.\ 98, 573--589 (1976; Zbl 0335.47018)] and the composition operator \(C_\varphi\) [see \textit{P.\,S.\thinspace Bourdon, J.\,A.\thinspace Cima} and \textit{A.\,L.\thinspace Matheson}, Trans.\ Am.\ Math.\ Soc.\ 351, No.\,6, 2183--2196 (1999; Zbl 0920.47029) and \textit{W.\,Smith}, Proc.\ Am.\ Math.\ Soc.\ 127, No.\,9, 2715--2725 (1999; Zbl 0921.47025)]. The boundedness of \(W_{\psi,\varphi}\) can be described by the facts that the two quantities \(\alpha(\psi,\varphi,a)=| \psi(a)| \| \sigma_{\varphi(a)}\circ \varphi\circ \sigma_a\| _{H^2}\) and \(\beta(\psi,\varphi,a)= (\log\frac{2}{1-| \varphi(a)| ^2})\| \psi\circ \sigma_a-\psi(a)\| _{H^2}\), where \(\sigma_a\) stands for the Möbious transform mapping \(\sigma(0)=a\), are bounded for \(| a| <1\). The proof is based on a weighted version of the Littlewood subordination principle. The author also studies the case \(VMOA\) and provides an asymptotic estimate for the essential norm of \(W_{\psi,\varphi}\), which seems to be new even in the case of multiplication and composition operators