We study the phenomenon of cospectrality in generalized line graphs and in exceptional graphs. We survey old results from today's point of view and obtain some new results partly by the use of computer. Among other things, we show that a connected generalized line graph $L(H)$ has an exceptional cospectral mate only if its root graph $H$, assuming it is itself connected, has at most 9 vertices. The paper contains a description of a table of sets of cospectral graphs with least eigenvalue at least $-2$ and at most 8 vertices together with some comments and theoretical explanations of the phenomena suggested by the table.