  In this paper, we study the explicit geometry of threefolds, in particular,
Fano varieties. We find an explicitly computable positive integer $N$, such
that all but a bounded family of Fano threefolds have $N$-complements. This
result has many applications on finding explicit bounds of algebraic invariants
for threefolds. We provide explicit lower bounds for the first gap of the
$\mathbb R$-complementary thresholds for threefolds, the first gap of the
global lc thresholds, the smallest minimal log discrepancy of exceptional
threefolds, and the volume of log threefolds with reduced boundary and ample
log canonical divisor. We also provide an explicit upper bound of the
anti-canonical volume of exceptional threefolds. While the bounds in this paper
may not and are not expected to be optimal, they are the first explicit bounds
of these invariants in dimension three.
