We present our new tools that are advancing network analysis towards a theoretical understanding of the structure of biological networks. Analogous to tools for analyzing and comparing genetic sequences, we are developing new tools that decipher large network data sets with the goal of improving biological understanding and contributing to development of new therapeutics. We demonstrate that local node similarity corresponds to similarity in biological function and involvement in disease. Also, we introduce a systematic, highly constraining measure of a network's local structure and demonstrate that protein-protein interaction (PPI) networks are better modeled by geometric graphs than by any previous model. The geometric model is further corroborated by demonstrating that PPI networks can explicitly be embedded into a low-dimensional geometric space and that evolutionary processes that constructed them can naturally be modelled in this space. We use these results to propose a new method for de-noising PPI data sets. Also, we present our new network alignment algorithms that are based only on network topology and are capable not only of function prediction, but also of reconstruction of phylogeny.