  We study the dependence of a cluster algebra on the choice of coefficients.
We write general formulas expressing the cluster variables in any cluster
algebra in terms of the initial data; these formulas involve a family of
polynomials associated with a particular choice of "principal" coefficients.
  We show that the exchange graph of a cluster algebra with principal
coefficients covers the exchange graph of any cluster algebra with the same
exchange matrix.
  We investigate two families of parametrizations of cluster monomials by
lattice points, determined, respectively, by the denominators of their Laurent
expansions and by certain multi-gradings in cluster algebras with principal
coefficients. The properties of these parametrizations, some proven and some
conjectural, suggest links to duality conjectures of V.Fock and A.Goncharov
[math.AG/0311245].
  The coefficient dynamics leads to a natural generalization of
Al.Zamolodchikov's Y-systems. We establish a Laurent phenomenon for such
Y-systems, previously known in finite type only, and sharpen the periodicity
result from [hep-th/0111053].
  For cluster algebras of finite type, we identify a canonical "universal"
choice of coefficients such that an arbitrary cluster algebra can be obtained
from the universal one (of the same type) by an appropriate specialization of
coefficients.
