Abstract: We discuss recent results towards the Lang-Trotter and Sato-Tate conjectures. While the Sato-Tate conjecture has recently been proved by Richard Taylor, in the case of the Lang-Trotter conjecture even the Extended Riemann Hypothesis is not powerful enough to establish the expected result.

We show that various techniques from analytic number theory help to establish new results for these conjectures on average over natural families of elliptic curve. These include the curves $Y^2 = X^3 + aX + b$ where $a$ and $b$ run through the intervals of the form $|a| \le A$, $b$ \le B and curves $Y^2 = X^3 + A(t)X + B(t)$, where $A$ and $B$ are fixed polynomials and $t$ runs through the set of Farey fractions of order $T$.

Some of the results are obtained jointly with Bill Banks (Univ. of Missouri, Columbia) and Alina Cojocaru (Univ. of Illinois, Chicago).