  For any separated algebraic space $X/S$ we construct a separated algebraic
space $\Gamma^d(X/S)$ -- the space of divided powers -- which parameterizes
zero cycles of degree $d$ on $X$. The space of divided powers for an affine
scheme is given by the spectrum of the algebra of divided powers. In
characteristic zero or when $X/S$ is flat, the constructed space coincides with
the symmetric product $Sym^d(X/S)$. We also prove several fundamental results
on the kernels of multiplicative polynomial laws necessary for the construction
of $\Gamma^d(X/S)$.
