  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 Luttinger-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 $\rm CuGeO_3$.

Strongly Correlated Electrons