The embedding of finite metrics in ℓ₁ has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems as there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into ℓ₁. Here we show that this theory can be generalized to a larger set of combinatorial optimization problems on both graphs and hypergraphs. This theory is not built on metrics and metric embeddings, but on diversities, a type of multi-way metric introduced recently by the authors. We explore diversity embeddings, ℓ₁ diversities, and their application to Steiner Tree Packing and Hypergraph Cut problems.