Bertini's Theorem over a general infinite field k states that given any smooth, quasi-projective variety X in P^n over k, there is at least one hyperplane in P^n over k whose scheme-theoretic intersection with X is smooth. This fails for k finite, but B. Poonen has shown, using sieve methods, that it can be salvaged if we allow higher degree hypersurfaces instead of just hyperplanes. He has also given a formula for the density of these "good" hypersurfaces. The first talk will discuss this theorem, its proof, and its consequences, and the second talk will discuss the (as yet conjectural) analog over Spec Z. In a special case, this analog reduces to the well-known fact that the density of the squarefree integers is 6/pi^2.