  We generalize the Bia{\l}ynicki-Birula decomposition from actions of $G_m$ on
smooth varieties to actions of linearly reductive group ${\bf G}$ on finite
type schemes and algebraic spaces. We also provide a relative version and
briefly discuss the case of algebraic stacks.
  We define the Bia{\l}ynicki-Birula decomposition functorially: for a fixed
${\bf G}$-scheme $X$ and a monoid $\overline{\bf G}$ which partially
compactifies ${\bf G}$, the BB decomposition parameterizes ${\bf G}$-schemes
over $X$ for which the ${\bf G}$-action extends to the $\overline{\bf
G}$-action. The freedom of choice of $\overline{\bf G}$ makes the theory richer
than the $G_m$-case.
