Previous works mostly focus on either multilingual or multi-domain aspects of neural machine translation (NMT). This paper investigates whether the domain information can be transferred across languages on the composition of multi-domain and multilingual NMT, particularly for the incomplete data condition where in-domain bitext is missing for some language pairs. Our results in the curated leave-one-domain-out experiments show that multi-domain multilingual (MDML) NMT can boost zero-shot translation performance up to +10 gains on BLEU, as well as aid the generalisation of multi-domain NMT to the missing domain. We also explore strategies for effective integration of multilingual and multi-domain NMT, including language and domain tag combination and auxiliary task training. We find that learning domain-aware representations and adding target-language tags to the encoder leads to effective MDML-NMT.