  After giving an overview of the existing theory regarding the periods of
sequences defined by linear recurrences over finite fields, we give explicit
descriptions of the sets of periods that arise if one considers all sequences
over $\mathbb{F}_q$ generated by linear recurrences for a fixed choice of the
degree $k$ in the range $1 \leq k \leq 4$. We also investigate the periods of
sequences generated by linear recurrences over rings of the form
$\mathbb{F}_{q_1} \oplus \ldots \oplus \mathbb{F}_{q_r}$.
