  In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue
$\lambda_{k}(G)$ of a graph $G$ of order $n$. This challenging question has
never been tackled for any $2<k<n$. In the present paper tight bounds are
obtained for all $k>2,$ and even tighter bounds are obtained for the $k$'th
largest singular value $\lambda_{k}^{\ast}(G).$
  Some of these bounds are based on Taylor's strongly regular graphs, and other
on a method of Kharaghani for constructing Hadamard matrices. The same kind of
constructions are applied to other open problems, like Nordhaus-Gaddum problems
of the kind: How large can $\lambda_{k}(G)+\lambda_{k}(\bar{G})$ be$?$
  These constructions are successful also in another open question: How large
can the Ky Fan norm $\lambda_{1}^{\ast}(G)+...+\lambda_{k}^{\ast }(G)$ be $?$
Ky Fan norms of graphs generalize the concept of graph energy, so this question
generalizes the problem for maximum energy graphs.
  In the final section, several results and problems are restated for
$(-1,1)$-matrices, which seem to provide a more natural ground for such
research than graphs.
  Many of the results in the paper are paired with open questions and problems
for further study.
