From Wikipedia:
Following the outbreak of World War I, Australian troops captured Kaiser-Wilhelmsland and the nearby islands in 1914, after a short resistance led by captain Carl von Klewitz and Lt. Robert "Lord Bob" von Blumenthal, while Japan occupied most of the remaining German possessions in the Pacific.
The only significant battle occurred on 11 September 1914, when the Australian Naval and Military Expeditionary Force attacked the low-power wireless station at Bita Paka (near Rabaul) on the island of New Britain, then Neu Pommern. The Australians suffered six dead and four wounded – the first Australian military casualties of the First World War. The German forces fared much worse, with one German officer and 30 native police killed and one German officer and ten native police wounded. On 21 September all German forces in the colony surrendered.
After the Treaty of Versailles of 1919, Germany lost all its colonial possessions, including German New Guinea. In 1923, the League of Nations gave Australia a trustee mandate over Nauru, with the United Kingdom and New Zealand as co-trustees. Other lands south of the equator became the Territory of New Guinea, a League of Nations Mandate Territory under Australian administration until 1949 (interrupted by Japanese occupation during the New Guinea campaign).
The islands north of the equator (the Marshall Islands, the Caroline Islands, the Marianas Islands, the Palau Islands, along with Kiautschou in China) became the Japanese League of Nations Mandate for the South Seas Islands. After Japan's defeat in World War II, the former Japanese mandate islands were administered by the United States as the Trust Territory of the Pacific Islands, a United Nations trust territory.