For smooth fibre bundles with fibre a closed connected oriented surface, there are certain characteristic classes, first studied by E. Miller, S. Morita, and D. Mumford, and therefore called Miller-Morita-Mumford (or MMM) classes; see, for example, Ch. 4 of Morita's book [\textit{S. Morita}, Geometry of characteristic classes. Transl. from the Japanese by the author. Translations of Mathematical Monographs. Iwanami Series in Modern Mathematics. 199. Providence, RI: American Mathematical Society (AMS). (2001; Zbl 0976.57026)]. The paper under review mainly studies a generalization of the MMM-classes to the context of fibre bundles with fibre a smooth closed manifold of arbitrary dimension. Let \(M\) be a smooth closed oriented \(n\)-dimensional manifold, and let \(\text{Diff}^ +(M)\) be the topological group of all orientation-preserving diffeomorphisms of \(M\). By a smooth oriented \(M\)-bundle (note that the adjective ``smooth''\, refers to \(M\), and not to the bundle), the author means a fibre bundle with structure group \(\text{Diff}^ +(M)\) and fibre \(M\); the base space of the bundle is not assumed to be a manifold. Let \(f: E\rightarrow B\) be a smooth oriented \(M\)-bundle, and \(P\rightarrow B\) be its associated \(\text{Diff}^ +(M)\)-principal bundle; \(E\) can be identified with \(P\times_{\text{Diff}^ +(M)} M\). Then the vertical tangent bundle of \(f\) is the oriented \(n\)-dimensional vector bundle \(T^ f:=P\times_{\text{Diff}^ +(M)} TM \rightarrow E\). For each rational cohomology class \(c\) of the classifying space \(BSO(n)\), one has the associated cohomology class \(c(T^ f)\in H^ *(E;\mathbb Q)\). The corresponding generalized Miller-Morita-Mumford or MMM-class of \(f\) is defined to be \({\kappa_ E}(c):=f_ {!}(c(T^ f)) \in H^ {*-n} (B;\mathbb Q)\), where \(f_ {!}: H^ *(E;\mathbb Q)\rightarrow H^ {*-n}(B;\mathbb Q)\) is the Gysin (or Umkehr) homomorphism. The central result of the paper under review, Theorem A, basically says that (1) if \(n\geq 0\) is even, then for each nonzero \(c\in H^ *(BSO(n);\mathbb Q)\), there is a connected smooth closed oriented \(n\)-dimensional manifold \(M\) and a smooth oriented \(M\)-bundle \(f: E\rightarrow B\) with \({\kappa_ E}(c)\neq 0\); (2) for \(n\geq 1\) odd, the analogous statement is ``almost true'' in the sense that there is one exception: the MMM-class associated with the Hirzebruch \(\mathcal L\)-class always vanishes. For \(n=2\), this theorem covers a known result for surface bundles that was first proved by \textit{E. Y. Miller} [J. Differ. Geom. 24, 1--14 (1986; Zbl 0618.57005)] and \textit{S. Morita} [Invent. Math. 90, 551--577 (1987; Zbl 0608.57020)], and the author of the paper under review uses this in his proof. A result similar to Theorem A is derived for holomorphic fibre bundles. In addition, the author explains how the results of this paper can be interpreted in terms of the Madsen-Tillmann-Weiss spectra, and he also deals with the case of unoriented-manifold bundles.