This paper proposes IPzip, a comprehensive suite of algorithms for compressing IP network packet headers and payloads. We propose an online algorithm for compressing packets in real-time for efficient transfer and an offline algorithm for efficient storage of the network data. In contrast to related approaches, IPzip achieves better compression by exploiting the correlations exhibited by (i) packets that are similar such as those belonging to the same layer-4 flow or those with the same destination port (inter-packet correlation) and (ii) header fields that are correlated to each other (intra-packet correlation). Since reordering of packets and fields is resource intensive, IPzip generates a near-optimal compression plan in an offline phase. Moreover, we propose a methodology to monitor over time the effectiveness of the compression plan being used and switch to a new compression plan when performance of the current compression plan decreases due to changes in the intrinsic traffic structure. Finally, via trace-driven experiments on network traffic obtained from Tier-1 ISPs, we validate that IPzip achieves better performance compared to related approaches.