  We study cube-free words over arbitrary non-unary finite alphabets and prove
the following structural property: for every pair $(u,v)$ of $d$-ary cube-free
words, if $u$ can be infinitely extended to the right and $v$ can be infinitely
extended to the left respecting the cube-freeness property, then there exists a
"transition" word $w$ over the same alphabet such that $uwv$ is cube free. The
crucial case is the case of the binary alphabet, analyzed in the central part
of the paper.
  The obtained "transition property", together with the developed technique,
allowed us to solve cube-free versions of three old open problems by Restivo
and Salemi. Besides, it has some further implications for combinatorics on
words; e.g., it implies the existence of infinite cube-free words of very big
subword (factor) complexity.
