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We generalize the Bia{\l}ynicki-Birula decomposition from actions of $G_m$ onsmooth varieties to actions of linearly reductive group ${\bf G}$ on finitetype schemes and algebraic spaces. We also provide a relative version andbriefly 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 partiallycompactifies ${\bf G}$, the BB decomposition parameterizes ${\bf G}$-schemesover $X$ for which the ${\bf G}$-action extends to the $\overline{\bfG}$-action. The freedom of choice of $\overline{\bf G}$ makes the theory richerthan the $G_m$-case.

Bia{\l}ynicki-Birula decomposition for reductive groups