a large part of this talk will be devoted to symmetric bilinear forms and inner product spaces, which should be of interest to geometers and algebraists alike. i'll outline some of the theory over general (commutative) rings and then turn to the classification of non-degenerate symmetric bilinear forms over the integers. the case of indefinite forms is completely understood and not too hard. on the contrary, the case of positive definite forms is quite difficult and, surprisingly, turns out to be related to the question of packing oranges in euclidean boxes. in the second part of the talk, we will see that symmetric bilinear forms come up as invariants in the theory of $4$-dimensional manifolds (their intersection forms). freedman showed that a (topological) $4$-manifold is (almost) classified by its associated bilinear form and that indeed every possible form arises in this way. on the other hand, donaldson showed that the intersection form of a \emph{smooth} $4$-manifold has a very special structure. combining the two results led to examples of many topological $4$-manifolds that do not admit a smooth structure, a revolution of the (previously almost non-existent) theory of $4$-manifolds, two fields medals, lots of other interesting research, and this talk.