  First, an algebraic criterion for integrability is discussed -the so-called
`superintegrability'- and some results on the classification of superintegrable
quantum spin Hamiltonians based on sl(2) are obtained.
  Next, the massive phases of the Z_n-chiral Potts quantum spin chain (a model
that violates parity) are studied in detail. It is shown that the excitation
spectrum of the massive high-temperature phase can be explained in terms of n-1
fundamental quasiparticles. We compute correlation functions from a
perturbative and numerical evaluation of the groundstate for the Z_3-chain. In
addition to an exponential decay we observe an oscillating contribution. The
oscillation length seems to be related to the asymmetry of the dispersion
relations. We show that this relation is exact at special values of the
parameters for general Z_n using a form factor expansion.
  Finally, we discuss several aspects of extended conformal algebras
(W-algebras). We observe an analogy between boundary conditions for Z_n-spin
chains and W-algebras and then turn to statements about the structure of
W-algebras. In particular, we briefly summarize results on unifying structures
present in the space of all quantum W-algebras.
