Summary: The interplay of charge, spin and lattice degrees of freedom is studied for quasi-one-dimensional electron and spin systems coupled to quantum phonons. Special emphasis is put on the influence of the lattice dynamics on the Peierls transition. Using exact diagonalization techniques the ground-state and spectral properties of the Holstein model of spinless fermions and of a frustrated Heisenberg model with magneto-elastic coupling are analyzed on finite chains. In the non-adiabatic regime a (T = 0) quantum phase transition from a gapless buttinger-liquid/spin-fluid state to a gapped dimerized phase occurs at a nonzero critical value of the electron/spin-phonon interaction. To study the nature of the spin-Peierls transition at finite temperatures for the infinite system, an alternative Green's function approach is applied to the magnetostrictive XY model. With increasing phonon frequency the structure factor shows a remarkable crossover from soft-mode to central-peak behaviour. The results are discussed in relation to recent experiments on CuGeO3.

From the random-phase approximation to the spin-Peierls transition, two parameter regimes of phonon softening and hardening are present. Magnetoelastic excitations are discussed in detail for phonons coupled to the exactly solvable model of XY spin chains for both regimes, leading to a modified interpretation of the 30-cm Ϫ1 mode in CuGeO 3 . Frustrated Heisenberg chains coupled to phonons satisfactorily describe the pretransitional quasielastic scattering in CuGeO 3 . A real-space interpretation of the quasielastic scattering is given justifying effective Ising-model approaches.

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