We investigate conditions for simultaneous normalizability of a family of reduced schemes; that is, the normalization of the total space normalizes, fiber by fiber, each member of the family. The main result (under more general conditions) is that a flat family of reduced equidimensional projective ${\mathbb C}$ -varieties $(X_y)_{y\in Y}$ with normal parameter space $Y$ —algebraic or analytic—admits a simultaneous normalization if and only if the Hilbert polynomial of the integral closure $\overline{{cal O}_{X_{y}}}$ is locally independent of $y$ . When the $X_y$ are curves, projectivity is not needed, and the statement reduces to the well-known $\delta$ -constant criterion ofTeissier. The proofs are basically algebraic, analytic results being related via standard techniques (Stein compacta and forth) to more abstract algebraic ones