In the past three decades there have been some exciting applications of elliptic curves over finite fields to integer factoring, primality testing, and cryptography. These applications in turn have raised some interesting problems often of an unconventional flavor. For example, how often is the order of an elliptic curve group prime, or how often does it have all small prime factors? In this talk we will visit problems such as these, as well as other analytic-type problems relating to ranks of elliptic curves over function fields and to elliptic divisibility sequences.