  We introduce the $k$-stellated spheres and consider the class ${\cal W}_k(d)$
of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its
subclass ${\cal W}^{\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members
of ${\cal W}_k(d)$. We introduce the mu-vector of any simplicial complex and
show that, in the case of 2-neighbourly simplicial complexes, the mu-vector
dominates the vector of its Betti numbers componentwise; the two vectors are
equal precisely for tight simplicial complexes. We are able to estimate/compute
certain alternating sums of the components of the mu-vector of any
2-neighbourly member of ${\cal W}_k(d)$ for $d\geq 2k$. As one consequence of
this theory, we prove a lower bound theorem for such triangulated manifolds, as
well as determine the integral homology type of members of ${\cal
W}^{\ast}_k(d)$ for $d \geq 2k+2$. As another application, we prove that, when
$d \neq 2k+1$, all members of ${\cal W}^{\ast}_k(d)$ are tight. We also
characterize the tight members of ${\cal W}^{\ast}_k(2k + 1)$ in terms of their
$k^{\rm th}$ Betti numbers. These results more or less answer a recent question
of Effenberger, and also provide a uniform and conceptual tightness proof for
all except two of the known tight triangulated manifolds.
  We also prove a lower bound theorem for triangulated manifolds in which the
members of ${\cal W}_1(d)$ provide the equality case. This generalises a result
(the $d=4$ case) due to Walkup and Kuehnel. As a consequence, it is shown that
every tight member of ${\cal W}_1(d)$ is strongly minimal, thus providing
substantial evidence in favour of a conjecture of Kuehnel and Lutz asserting
that tight triangulated manifolds should be strongly minimal.
