Bathroom GFCI receptacle protection is required by the NEC (National Electrical Code), and of course by the common sense - why would you risk getting electrocuted if you can secure your safety by installing GFCI outlet? Since some bathrooms are currently equipped with many electrical current hungry devices, power supply requirements had to be slightly adjusted.
We used to have a bathroom sharing an electrical circuit with exterior outlets, the kitchen, and a garage - not anymore.
Of course, the minimum requirement listed below is not going to work if you add a steam shower, heated electric floor, hydro-spa heater, several light fixtures, heat lamps, etc.
For all that additional equipment load calculation would have to be performed, but now, let me cover the basics.
The minimum requirement for a single bathroom is one, 20-ampere rated electrical circuit.
This electrical circuit should not be used for anything else than outlets and equipment within the bathroom.
All new bathroom outlet receptacles must be GFCI protected, and 20 ampere rated (assuming, they are installed on #12 wire).

• Each bathroom sink must have a GFCI receptacle located within 3' from its outside edge
• If you are replacing an old, not GFCI type bathroom receptacle, you have to install currently required GFCI type outlet in its place (unless this circuit is already protected by another GFCI receptacle or GFCI breaker - however a breaker will not protect ungrounded electrical installation)
• Hydro-massage tub must be protected by a GFCI receptacle or breaker, and must have a disconnecting means in sight of motor - most of the tubs have a cord and plug which can be simply pulled out of the socket.
  Hydro-spa with a hard wired motor should have a ON/OFF switch mounted in sight of motor
• Hydro-massage tub motor terminal needs to be bonded to the water pipe (cold) with solid #8 copper wire.
  The bonding is prohibited for a listed double-insulated type motors (it should say on the motor label, and there will be no terminal).
  
• Hydro-massage tub motor must be accessible, and the access panel large enough for servicing.
  

Bathroom electrical components Bathroom Lighting: There are some types of light fixtures listed below that are not permitted within a certain area in the bathroom.
This area is located directly above the bathtub and the shower stall, and additionally extends 3ft (0.
9 meter) horizontally, and 8ft (2.
5 meter) vertically from the bathtub top rim and the shower stall threshold.
The forbidden types of electrical light fixtures or parts of fixtures in this area are:

• Cord connected lights (i.
  e.
  table lamp)
• Chain / cable or a suspended cord
• Track type
• Pendants
• Ceiling fans

Any electrical light fixture located within that area must be marked for damp locations or wet locations where it's subject to shower spray.

• Bathroom light fixtures electrical wiring must be concealed within the electrical junction box or fixture enclosure After seeing peoples' lack of common sense in many hazardous situations, I always recommend to put any of the light fixtures located above the bathtub and / or shower stall on a GFCI protected circuit...
  but that's your choice (it is not required by code but may be your local jurisdiction and / or light fixture manufacturer requirement).
  

Bathroom Exhaust Fan: one last electrical component which under some conditions would require bathroom GFCI protection NEC (National Electrical Code) does not require bathroom exhaust fans to be installed on a GFCI protected circuit.
However, you should be following the manufacturer's installation instructions.
In most, or should I say in all, cases concerning a bathroom exhaust fan installation directly above the bathtub and / or shower stall (or any other moisture sources), the bathroom fan must be listed as suitable for such locations and must be installed on a GFCI protected circuit.
This is a paragraph from the bathroom exhaust fan installation instructions: "If this bathroom exhaust fan is to be installed over a tub or shower, it must be marked as appropriate for the application and be connected to a GFCI (Ground Fault Circuit Interrupter) protected branch circuit.
" To test if a bathroom exhaust fan is GFCI protected, turn the fan ON, and push TEST button on a bathroom GFCI receptacle or GFCI breaker dedicated for this bathroom.
If the exhaust fan is still running after the receptacle or the breaker tripped, it is most likely not protected (there are occasionally some hidden GFCI receptacles installed).
Final pledge - please, no electrical panels in the bathroom, and remember that properly installed bathroom GFCI could save your life.