A good description of the technical work performed by IANA is captured in RFC 2860. IANA was run by Jonathan Bruce Postel for thirty years until he passed away in 1998.
Due to the Internet growth during the 1990's there was also the need for an organization that would take responsibility over the central registration of domain names and addresses.
This is how ICANN (Internet Corporation for Assigned Names and Numbers) was created.
IANA is responsible for the delegation of IP Addresses to RIRs.
In their turn, each RIR (Regional Internet Registry) is responsible for the allocation of IP addresses depending on the zone/area they are managing.