Cats and dogs are accepted by nearly every country if you follow the import protocol; the paperwork, not the principle, is the challenge. The usual restrictions worth checking first: several countries ban or heavily restrict specific dog breeds, typically those classified as fighting or dangerous breeds, and a banned breed can be refused entry entirely regardless of temperament or paperwork.
Birds, reptiles, rabbits, and exotic pets are a different world: many require import permits, quarantine, or are prohibited, and anything covered by endangered species rules needs CITES documentation. Airlines add their own layer, with breed embargoes for snub-nosed dogs on some carriers and seasonal temperature restrictions for animals in cargo.
Start planning months ahead, because vaccination and testing timelines cannot be compressed. The authoritative source is the destination country's agriculture or veterinary authority website, which publishes the exact import conditions by species and origin country. A specialist pet relocation agent is worth considering for complex routes or restricted breeds.