While it is generally safe to travel while pregnant, there are several guidelines, restrictions, and safety measures of which you'll want to be aware. Keep in mind, all of the information here pertains specifically to flights which are operated by Air New Zealand; different airlines, even those who are partners of Air New Zealand, will likely have their requirements that you should check before booking.
Depending on how far along the pregnancy is, Air New Zealand follows strict regulations in regard to how long and when a pregnant woman can fly. These regulations vary depending on whether your pregnancy is a single pregnancy or multiple pregnancies (i.e., twins or triplets).
For a single baby pregnancy, the pregnant passenger may board flights over four hours until the end of the 36th week. Flights less than four hours may be boarded up until the close of the 40th week. For women carrying twins or triplets, flights of four hours or more are permitted until the end of the 32nd week. Those under four hours are approved until the 36th week.
Keep in mind that all women wishing to fly while carrying multiple pregnancies are asked for clearance from either a doctor a midwife. If the pregnancy is considered "complicated" in any way, you are not advised to travel.
In some cases, women wishing to travel while pregnant will be required to receive approval from the medical team in order to be cleared for a flight. For example if you are in the early stages of labour, if you have a medical history of premature labour, if you are carrying twins or triplets, and if your pregnancy is considered complicated in nature, you'll be required to consult with the Air New Zealand's Aviation Medical team in order to be cleared for travel. In most cases, assessments are quick and straightforward.
In any case, women who have passed their 28th week of pregnancy are recommended to obtain a letter from a doctor or a midwife stating explicitly that the pregnant passenger is cleared for travel. Ideally, the letter should also confirm the pregnancy dates and state that the pregnancy is low-risk with zero known complications. Just in case, it's best to carry this letter with you at all times as you travel.
As a rule of thumb, Air New Zealand recommends always booking international trips long before your baby is due. Because pregnancy and the stress that comes with it can add additional complications to your holiday plans, it's also heavily recommended that you consider covering your trip with travel insurance.
Before booking flights with any airline, check the entry restrictions for pregnant foreigners. Some countries have rules and regulations which restrict the travel of pregnant non-citizens into the country. This can be done at the appropriate embassy or consulate nearest to you.
Once you've been cleared for travel and board the plane, you'll want to be proactive about maintaining your wellbeing during the flight. This includes drinking plenty of water, elevating your legs, and going for short walks when possible.
To promote circulation and ease the strain on muscles, Air New Zealand also recommends that pregnant passengers stay conscious about seated posture, regularly perform small flexing and muscle rotations, and refrain from crossing their legs.
Source: Air New Zealand