According to CIN and their 2013 Cruise Industry News Annual Report, they state that with the delivery of MSC Preziosa, the line will over take NCL as the #3 line. The line will carry over 1.45 million passengers on their 12 ships. NCL also has 12 ships, however MSC vessels carry more pax.
Princess has 17 ships currently, however they deploy their ships on longer routes, so their pax count is down compared to MSC.
CIN reports that Carnival and Royal Caribbean each carry about 3 million each.