On normal fares your baggage allowance is either weight or piece concept. Itineraries involving Nth America should come under the piece system, whereas the weight system operates elsewhere.
All *RTW's operate under the piece system - 2 pieces per person with a maximum weight limit of 32kg or 70lb per piece.

FWIW, the official harmonised * Baggage allowances are as follows:
Piece Concept -
2 pcs per person in F J and Y max weight 32kg/70lb
Infants not occupying a seat - 1 piece at same weight restriction
*G members get one additional free piece at same weight levels.

Weight Concept - unlimited pieces within a maximum combined weight limit of ........
F get 40kg/88lb
J are allowed 30kg/66lb
Y get 20kg/44lbs
Infants not occupying a seat get 10kg/22lb
*G members receive an extra 20kg/44lb allowance.

Carry-On Baggage -
*'s official policy is a 2x2x1 - 2 in F, 2 in J and 1 in Y. Dimensions are capped at a maximum of 115cm (L+W+H)
Carry-on policy is for Intercontinental flights and Domestic flights connecting to Intercontinental flights. Each carrier still has it's own policy on Domestic travel.

As you mentioned, some carriers may have extra pieces allowed under their own rules and class of travel, but this would apply mainly to normal fares, more so if travel is wholly on that carrier.

Additionally, there is a harmonised policy on "extra bits". Bicycles, for example, are meant to be carried free if they are within the baggage allowance limits. A set of Golf Clubs is generally accepted free even if that set takes one over the limit allowed. Other sporting gear such as surfboards and ski sets are also covered, however I believe these are forgotten by many carriers.

Cheers.