  This paper initiates the study of circular orderability of $3$-manifold
groups, motivated by the L-space conjecture. We show that a compact, connected,
$\mathbb{P}^2$-irreducible $3$-manifold has a circularly orderable fundamental
group if and only if there exists a finite cyclic cover with left-orderable
fundamental group, which naturally leads to a "circular orderability version"
of the L-space conjecture. We also show that the fundamental groups of almost
all graph manifolds are circularly orderable, and contrast the behaviour of
circularly orderability and left-orderability with respect to the operations of
Dehn surgery and taking cyclic branched covers.
