  Regarding neighbor balance, we consider natural generalizations of
$D$-complete Latin squares and Vatican squares from the finite to the infinite.
We show that if $G$ is an infinite abelian group with $|G|$-many square
elements, then it is possible to permute the rows and columns of the Cayley
table to create an infinite Vatican square. We also construct a Vatican square
of any given infinite order that is not obtainable by permuting the rows and
columns of a Cayley table. Regarding orthogonality, we show that every infinite
group $G$ has a set of $|G|$ mutually orthogonal orthomorphisms and hence there
is a set of $|G|$ mutually orthogonal Latin squares based on $G$. We show that
an infinite group $G$ with $|G|$-many square elements has a strong complete
mapping; and, with some possible exceptions, infinite abelian groups have a
strong complete mapping.
