In this talk we shall be concerned with the residues modulo $4$ and modulo $8$ of the signature $\sigma(M) \in \mathbb{Z}$ of an oriented 4k-dimensional geometric Poincare complex $M$. The precise relation between the signature modulo $8$ and the Brown-Kervaire invariant was worked out by Morita. We shall discuss how the relation between these invariants and the Arf invariant can be applied to the study of the signature modulo $8$ of a fibration. In particular it had been proved by Meyer in 1973 that a surface bundle has signature divisible by $4$. This was generalized to higher dimensions by Hambleton, Korzeniewski and Ranicki in 2007. I will explain two results from my thesis concerning the signature modulo $8$ of a fibration: firstly under what conditions can we guarantee divisibility of the signature by 8 and secondly what invariant detects non-divisibility by 8 in general. I will also talk about a joint project with S. Yokura where we investigate how multiplicativity properties of the signature modulo $8$ also hold for certain Hirzebruch $\chi_y$-genera.