A Kiribati man is seeking asylum because of environmental changes that have caused the sea level to rise.
He has been refused refugee status by Immigration New Zealand and the Immigration Protection Tribunal, and is seeking leave to appeal.
Ioane Teitiota says there is no land anywhere in Kiribati for his family to relocate to because of climate change.
The Immigration Protection Tribunal ruled that, while the claims are credible and accepted, they do not meet the criteria of the Refugee Convention.
Mr Teitiota argues the tribunal failed to consider the serious harm his three New Zealand-born children would suffer if they were deported.
His lawyer told the court the family is concerned for their health and well-being if they return to Kiribati as land is being eaten away by sea-level rise, water is contaminated by salt and sewerage, and crops are dying.
He said the case shows how outdated the Refugee Convention is.
The judge has reserved his decision, expected in the next two weeks.