What are the roles of the HLR, VLR, AUC, and GMSC in the UMTS core network?

Question

Accepted Answer

HLR = permanent subscriber master database; VLR = temporary database for visitors in an MSC's area; AUC = authentication/key center; GMSC = the gateway that connects the operator's network to external telephone networks.
These subscriber-management and gateway elements were carried over almost unchanged from GSM into the UMTS core:

Component
Full name
Role

HLR
Home Location Register
The master database of every subscriber registered with the operator — stores fundamental user data: identity, subscribed services, tariff, and the user's current location area. One per operator (logically).

VLR
Visitor Location Register
A temporary copy of data for subscribers currently in a particular MSC's service area. Pulls data from the HLR when a phone enters, so the MSC doesn't have to query the HLR for every action.

AUC
Authentication Center
Securely stores the secret keys and generates the authentication vectors used to verify subscribers (and, in UMTS, to let the network prove itself back to the phone). Usually co-located with the HLR.

GMSC
Gateway MSC
The interface between the operator's own network (UMTS-PLMN) and external networks such as the fixed PSTN/ISDN telephone network. An incoming call from the outside world first hits the GMSC, which asks the HLR where the subscriber currently is.

HLR vs VLR mental model: The HLR is the permanent home record ("where you're registered"); the VLR is a temporary local cache ("who is visiting this MSC right now"). The GMSC is the front door between this operator and the rest of the world's phone networks.
Go deeper:

3G UMTS Network Architecture (Electronics Notes) — describes the HLR/VLR/AUC/EIR shared core elements and the GMSC gateway, and how an incoming call hits the GMSC, which queries the HLR to locate the subscriber.

Component	Full name	Role
HLR	Home Location Register	The master database of every subscriber registered with the operator — stores fundamental user data: identity, subscribed services, tariff, and the user's current location area. One per operator (logically).
VLR	Visitor Location Register	A temporary copy of data for subscribers currently in a particular MSC's service area. Pulls data from the HLR when a phone enters, so the MSC doesn't have to query the HLR for every action.
AUC	Authentication Center	Securely stores the secret keys and generates the authentication vectors used to verify subscribers (and, in UMTS, to let the network prove itself back to the phone). Usually co-located with the HLR.
GMSC	Gateway MSC	The interface between the operator's own network (UMTS-PLMN) and external networks such as the fixed PSTN/ISDN telephone network. An incoming call from the outside world first hits the GMSC, which asks the HLR where the subscriber currently is.