  We introduce the class $\Sigma_k(d)$ of $k$-stellated (combinatorial) spheres
of dimension $d$ ($0 \leq k \leq d + 1$) and compare and contrast it with the
class ${\cal S}_k(d)$ ($0 \leq k \leq d$) of $k$-stacked homology $d$-spheres.
We have $\Sigma_1(d) = {\cal S}_1(d)$, and $\Sigma_k(d) \subseteq {\cal
S}_k(d)$ for $d \geq 2k - 1$. However, for each $k \geq 2$ there are
$k$-stacked spheres which are not $k$-stellated. The existence of $k$-stellated
spheres which are not $k$-stacked remains an open question.
  We also consider the class ${\cal W}_k(d)$ (and ${\cal K}_k(d)$) of
simplicial complexes all whose vertex-links belong to $\Sigma_k(d - 1)$
(respectively, ${\cal S}_k(d - 1)$). Thus, ${\cal W}_k(d) \subseteq {\cal
K}_k(d)$ for $d \geq 2k$, while ${\cal W}_1(d) = {\cal K}_1(d)$. Let
$\bar{{\cal K}}_k(d)$ denote the class of $d$-dimensional complexes all whose
vertex-links are $k$-stacked balls. We show that for $d\geq 2k + 2$, there is a
natural bijection $M \mapsto \bar{M}$ from ${\cal K}_k(d)$ onto $\bar{{\cal
K}}_k(d + 1)$ which is the inverse to the boundary map $\partial \colon
\bar{{\cal K}}_k(d + 1) \to {\cal K}_k(d)$.
