  The main theorem of this paper is that, for a general pair $(A,X)$ of an
(ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $\Phi_X$
of $X$ is birational onto its image if the polarization given by $X$ is not
principal (i.e., its Pfaffian $d$ is not equal to $1$).
  We also show that, setting $g = dim (A)$, and letting $d$ be the Pfaffian of
the polarization given by $X$, then if $X$ is smooth and $$\Phi_X : X
\rightarrow \mathbb{P}^{N:=g+d-2}$$ is an embedding, then necessarily we have
the inequality $ d \geq g + 1$, equivalent to $N : = g+d-2 \geq 2 \ dim(X) +
1.$
  We also formulate the following interesting conjecture, motivated by work of
the second author: if $ d \geq g + 1,$ then, for a general pair $(A,X)$,
$\Phi_X$ is an embedding.
