Weil conjectures - motivation? Can anyone explain (heuristically, intuitively is fine) what the importance of the Weil conjectures is? I realize they have motivated much of recent algebraic geometry. I don't really understand why or what one should do with them, now that they have been proven.

One very incredible thing about the Weil Conjectures is what they say about the geometry of a smooth, projective variety $X$ defined over $\mathbb{Q}$ (for example); some of the geometry of $X$ as a complex manifold can be computed by counting points on $X$ modulo $p$ for some (suitable) prime.  Specifically, the Weil Conjectures allow you to compute the Betti numbers (dimensions of vector spaces coming from cohomology) of $X/\mathbb{C}$ by computing the Zeta function of $X/\mathbb{F}_p$.  This Zeta function has properties quite analogous to the Riemann Zeta function, including a Riemann Hypothesis.  It is a rational function of a single variable, and it is constructed with a(n exponential of a) power series, the $n$th coefficient being the number of points on $X$ over the finite field $\mathbb{F}_{q}$, for $q=p^n$.  If you're familiar with Elliptic Curves, the Hasse-Weil inequality
$$
\left| \#E(\mathbb{F}_q) -(q+1) \right| \leq 2\sqrt{q}
$$
for an elliptic curve $E/\mathbb{F}_q$, is secretly the Riemann Hypothesis in disguise.
