Let E/Q be an elliptic curves without Complex Multiplication, and let K be an imaginary quadratic field. For a prime p, let pi_p be the Frobenius of E modulo. We will discuss results related to a conjecture by Lang and Trotter from 1976 on the number of primes p < x for which the field Q(pi_p) equals K. We will also discuss the analogous question for Drinfeld modules of rank 2. (Joint work with Chantal David.)