  A positive space is a space with a positive atlas, i.e. a collection of
rational coordinate systems with subtraction free transition functions. The set
of positive real points of a positive space is well defined. We define a
tropical compactification of the latter. We show that it generalizes the
Thurston compactification of a Teichmuller space. The tropical boundary of a
positive space is a sphere with a piecewise linear structure. Cluster
X-varieties are positive spaces of rather special type. We define special
completions of cluster X-varieties. They have a stratification whose strata are
(affine closures of) cluster X-varieties. The original coordinate tori extend
to coordinate affine spaces in the completion. We define completions of
Teichmuller spaces for surfaces with marked points at the boundary. The set of
positive points of the special completion of the corresponding cluster
X-variety is a part of the completion of the Teichmuller space.
