Let F = {F_1, F_2,..., F_n} be a family of n sets on a ground set X, such as a family of balls in R^d. For every finite measure \mu on X, such that the sets of F are measurable, the classical inclusion-exclusion formula asserts that \mu(F_1 \cup F_2 \cup \bullet \bullet \bullet \cup F_n) = \sum_{I:{\O} \neq I\subseteq[n]} (-1)^{|I|+1} \mu(\cap_{i\inI} F_i); that is, the measure of the union is expressed using measures of various intersections. The number of terms in this formula is exponential in n, and a significant amount of research, originating in applied areas, has been devoted to constructing simpler formulas for particular families F. We provide the apparently first upper bound valid for an arbitrary F: we show that every system F of n sets with m nonempty fields in the Venn diagram admits an inclusion- exclusion formula with m^O((log n)^2) terms and with \pm1 coefficients, and that such a formula can be computed in m^O((log n)^2) expected time. We also construct systems of n sets on n points for which every valid inclusion-exclusion formula has the sum of absolute values of the coefficients at least \Omega(n^{3/2}).