  Let $S$ be a subset of a group $G$ (not necessarily abelian) such that
$S\,\cap -S$ is empty or contains only elements of order $2$, and let
$\mathbf{h}=(h_1,\ldots, h_m)\in \mathbb{N}^m$ and $\mathbf{k}=(k_1, \ldots,
k_n)\in \mathbb{N}^n$. A generalized Heffter array GHA$^{\lambda}_S(m, n;
\mathbf{h}, \mathbf{k})$ over $G$ is an $m\times n$ matrix $A=(a_{ij})$ such
that: the $i$-th row (resp. $j$-th column) of $A$ contains exactly $h_i$ (resp.
$k_j$) nonzero elements, and the list $\{a_{ij}, -a_{ij}\mid a_{ij}\neq 0\}$
equals $\lambda$ times the set $S\,\cup\, -S$. We speak of a zero sum (resp.
nonzero sum) GHA if each row and each column of $A$ sums to zero (resp. a
nonzero element), with respect to some ordering.
  In this paper, we use near alternating sign matrices to build both zero and
nonzero sum GHAs, over cyclic groups, having the further strong property of
being simple. In particular, we construct zero sum and simple GHAs whose row
and column weights are congruent to $0$ modulo $4$. This result also provides
the first infinite family of simple (classic) Heffter arrays to be rectangular
($m\neq n$) and with less than $n$ nonzero entries in each row. Furthermore, we
build nonzero sum GHA$^{\lambda}_S(m, n; \mathbf{h}, \mathbf{k})$ over an
arbitrary group $G$ whenever $S$ contains enough noninvolutions, thus extending
previous nonconstructive results where $\pm S = G\setminus H$ for some subgroup
$H$~of~$G$.
  Finally, we describe how GHAs can be used to build orthogonal decompositions
and biembeddings of Cayley graphs (over groups not necessarily abelian) onto
orientable surfaces.
