How many domains are there?

There are literally hundreds of millions of active domains. 

Then there are many millions which are inactive, i.e. they were purchased and registered but are not hosted actively at the moment, i.e. having their own unique website. 

Most of domains have .com, .net extensions but there are also many millions having top country level domains, like .de, .fr, .es, and so on

One interesting approach of finding these domains is by looking at open source data sets. There are many out there, some with 90 millions of domains.