The tools of algebraic topology, such persistent homologies, or analysis of properties of progressively finer simplical complexes, allow to unveil some critical characteristics behind organization of complex networks and interactions of their components at multi-scale levels, which are otherwise largely unaccessible with conventional analytical approaches. We discuss utility of persistent homologies in amplifying our understanding of hidden mechanisms behind functionality and robustness of complex weighted time-evolving networks in a broad range of real-world scenarios.