There are many airlines that fly to the island of Roatan which is the biggest of the Bay Islands and the only one that has an international airport as of today. Here are the airlines that fly to and from Roatan in 2019.
From Canada (Montreal and Toronto): TRANSAT.
From Italy (Milano): Jolly Roger Tour Italy.
From the United States: Continental Airlines, Delta Airlines, American Airlines, Cayman Airways.
From Cayman Islands (Cayman Brac): Cayman Airways. This one also takes you to Cuba, Jamaica and the United States (Denver, Tampa, Miami, New York).
From Belize (San Pedro): Tropic Air Belize.
From Latin America (Central and South): AVIANCA, CM Airlines.
From the mainland Honduras: AVIATSA (charter planes), CM Airlines, AEROSOSA, LANHSA, AEROCARIBE.