  We obtain distribution results for traces of Frobenius for various families
of elliptic curves with respect to the Lang-Trotter conjecture, extremal
primes, and the central limit theorem. This includes some generalisations and
bounds related to the work of Sha-Shparlinski on the average Lang-Trotter
conjecture for single-parametric families of elliptic curves and the work of
various authors on the trace of Frobenius for primes in congruence classes.
Some results are also obtained for modular forms.
