There are several different religions claiming to be the "fastest growing religion". Such claims vary because of different definitions of "fastest growing", and whether the claim is worldwide or regional.
There are also many unreliable claims and rumors, especially for conversion rates, that often spread as urban legends.
Different definitions of "fastest growing"
Religions can grow in numbers because of conversion or because of higher birth rates in a religious group (assuming that children take on the religion of their parents).
Religions in particular countries can grow because of immigration.
So the fastest growing religion could refer to:
The religion whose absolute number of adherents is growing the fastest (by whatever means).
The religion that is growing fastest in terms of percentage growth per year (by whatever means).
The religion that is gaining the greatest number of converts.
The religion that is gaining the greatest number of associative members (those associating themselves via survey, effectively a popularity vote)
Measures counting absolute numbers tend to favour the larger religions; measures counting percentage growth the smaller ones.
For example, if a religion had only 10 followers, a single addition would be a 10% increase, and would therefore dwarf the percentage growth rates of the larger religions.
The difficulty of gathering data
Statistics on religious adherence are difficult to gather and often contradictory; statistics for the change of religious adherence are even more so, requiring multiple surveys separated by many years
using the same data gathering rules.
This has only been achieved in rare cases, and then only for a particular country, such as the American Religious Identification Survey in the USA, or census data from Australia
(which has included a voluntary religious question since 1911).
Worldwide data are more difficult to gather than data on a particular country.
Statistics for rates of conversion are the most difficult to gather and the least reliable: they are often distorted by social taboos such as the ban on apostasy in Islam, sometimes amplified by governments and policies
at social institutions like universities or the reporting of commitments where the individual does not persist.
This means that a lot of the data on growth of religions is derived from birth and immigration rates.
There are a large number of people who self-identify themselves as associated to a specific religion, but who are not religiously active.
If, for example, asked to choose between Christianity and other religions they would say they were Christians; if asked to choose between Christianity, other religions and "Not religious", they would say "Not religious".
This may make categorization difficult.
In countries with mandatory religions, official statistics will only reflect the official position of the government.