The Lang-Trotter Conjecture on primitive points is the analogue for elliptic curves of Artin's conjecture on primitive roots. Indeed, if we have an elliptic curve $E$ over $\mathbb Q$ with a rational point $P$ of infinite order, we may count the primes $p$ of good reduction for which $(P \bmod p)$ generates $E(\mathbb F_p)$. We formulate and investigate some natural variants of the Lang-Trotter Conjecture. For example, we require that the order of the point $(P \bmod p)$ equals the exponent of the group $E(\mathbb F_p)$: this means that the subgroup generated by the point is as large as possible, and the condition is meaningful also for non-cyclic groups. This is joint work with Alexandre Benoist.