  A linked system of symmetric designs (LSSD) is a $w$-partite graph ($w\geq
2$) where the incidence between any two parts corresponds to a symmetric design
and the designs arising from three parts are related. The original construction
for LSSDs by Goethals used Kerdock sets, in which $v$ is a power of two. Some
four decades later, new examples were given by Davis et.\ al.\ and Jedwab et.\
al.\ using difference sets, again with $v$ a power of two. In this paper we
develop a connection between LSSDs and "linked simplices", full-dimensional
regular simplices with two possible inner products between vertices of distinct
simplices. We then use this geometric connection to construct sets of
equiangular lines and to find an equivalence between regular unbiased Hadamard
matrices and certain LSSDs with Menon parameters. We then construct examples of
non-trivial LSSDs in which $w$ can be made arbitarily large for fixed even part
of $v$. Finally we survey the known infinite families of symmetric designs and
show, using basic number theoretic conditions, that $w=2$ in most cases.
