Methane
Methane is a powerful but relatively short lived greenhouse gas.  Averaged over 100 years, the global warming potential of one tonne of methane is 25 times that of one tonne of carbon dioxide. What is Global Warming Potential?
In New Zealand, most methane comes from belching by ruminant animals: sheep, beef cattle, dairy cows, and deer, and some from animal manure. Non-ruminants, such as poutry and pigs also produce methane from their manure but these constitute a very small source in New Zealand.
Nitrous oxide
Nitrous oxide is a much more powerful greenhouse gas than methane - 298 times more powerful than carbon dioxide over a 100 year timeframe on a per tonne basis.  See What is Global Warming Potential?
Fortunately, the absolute amount of nitrous oxide emitted from agriculture is lower than methane. In New Zealand, most nitrous oxide is produced by the action of soil bacteria in urine patches in paddocks.  Smaller amounts come from dung deposited during grazing, stored manures spread back onto pasture, and from nitrogen fertilizer. 
Soil carbon
Soil is an important reservoir of carbon which means that relatively small changes in the amount of carbon stored in soil could have significant effects on net greenhouse gas emissions.
Management approaches that increase the amount of carbon stored in soils could offset some of the emissions of other greenhouse gases from agriculture, methane and nitrous oxide.
On the other hand, practices that deplete soil carbon and release it back into the atmosphere can add to the emissions of other greenhouse gases. From a climate change perspective, what matters is not the total quantity of carbon in soils, but whether this quantity changes over time.
Given the potentially significant contribution that changes in soil carbon could make to New Zealand’s total agricultural emissions, it is essential to understand trends and identify management practices that can increase soil carbon – but equally, to avoid practices that would result in carbon losses.