  In this paper we study pluricanonical maps of minimal projective 3-folds of
general type with geometric genus $1$, $2$ and $3$. We go in the direction
pioneered by Enriques and Bombieri, and other authors, pinning down, for low
projective genus, a finite list of exceptions to the birationality of some
pluricanonical map. In particular, apart from a finite list of weighted
baskets, we prove the birationality of $\varphi_{16}$, $\varphi_{6}$ and
$\varphi_{5}$ respectively.
