A many-one reduction of problem $A$ to problem $B$ is essentially a function that converts a problem instance in problem $A$ to an instance in $B$. This allows you to use a $B-$solver one time to solve this converted instance, in order to solve an $A$-instance.
To show that $A$ is in NP-complete, people use reducibillity arguments to reduce (in polynomial time) an arbitrary problem in NP to $A$, but as far as I know this is not necessarily a many-one reduction.
My question is: How many of these reductions in the context of NP-completeness are many-one reductions? How significantly would the complexity class change if we would require many-one reductions?