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Cacomantis is a genus of cuckoo in the family Cuculidae. 

1/ Plaintive cuckoo (Tìm vịt, Cacomantis merulinus)
2/ Banded bay cuckoo or bay-banded cuckoo (Tìm vịt vằn, Cacomantis sonneratii)

The genus name is from the Greek kakos meaning evil or ill-boding and mantis for prophet. 
The name is associated with or supposed to be predicted "rains" as well as ill fortune and bad weather.

Carpococcyx is a genus of large terrestrial cuckoos in the family Cuculidae.

1/ Coral-billed ground cuckoo (Phướn đất, Carpococcyx renauldi), also known as Renauld's ground cuckoo

A coucal is one of about 30 species of birds in the cuckoo family. All of them belong in the subfamily Centropodinae and the genus Centropus. 
The genus name combines the Ancient Greek: kentron meaning "spur" or "spike", pous meaning "foot".

1/ Greater coucal or crow pheasant (Bìm bịp lớn, Centropus sinensis)
2/ Lesser coucal (Bìm bịp nhỏ, Centropus bengalensis) 

Chrysococcyx (formerly Chalcites) is a genus of cuckoo in the family Cuculidae. The word Chrysococcyx combines the Ancient Greek khrusos meaning "gold" and kokkux "cuckoo".

1/ Violet cuckoo (Tìm vịt tím, Chrysococcyx xanthorhynchus) 
2/ Little bronze cuckoo (Tìm vịt trán trắng, Chrysococcyx minutillus)
3/ Asian emerald cuckoo (Tìm vịt xanh, Chrysococcyx maculatus) 
The specific epithet maculatus is from Latin and means "spotted" or "blotched".

Clamator is a genus of large brood-parasitic cuckoos with crests and graduated tails. The name Clamator is Latin for "shouter" from clamare, "to shout".

1/ Chestnut-winged cuckoo or red-winged crested cuckoo (Khát nước, Clamator coromandus)

Cuculus is a genus of cuckoos which has representatives in most of the Old World with the greatest diversity in tropical southern and southeastern Asia. The genus name is the Latin word for "cuckoo".

1/ Indian cuckoo ( Bắt cô trói cột, Cuculus micropterus)
2/ Common cuckoo ( Cu cu, Cuculus canorus)
3/ Lesser cuckoo ( Cu cu nhỏ, Cuculus poliocephalus) 
4/ Himalayan cuckoo (Cu cu Himalaya, Cuculus saturatus)
5/ Oriental cuckoo or Horsfields cuckoo (Cu cu phương đông, Cuculus optatus), formerly classified as a subspecies of the Himalayan cuckoo (Cuculus saturatus)

The true koels, Eudynamys, are a genus of cuckoos from Asia, Australia and the Pacific. The name combines the Ancient Greek eu meaning "fine" with dunamis meaning "power" or "strength".

1/ Asian koel (Tu hú, Eudynamys scolopaceus)
The specific epithet scolopaceus is Modern Latin meaning "snipe-like" from Latin scolopax meaning "snipe" or "woodcock".

Phaenicophaeus is a genus of 6 species of cuckoos in the family Cuculidae. The birds are mostly known as Malkoha. 
The English name is derived from the Sinhala word for the red-faced malkoha; mal-koha meaning flower-cuckoo. 
The genus name is derived from the Ancient Greek: phoiniko- crimson, and phaes meaning "eyes" or "face", referring to the red-faced malkoha. 

1/ Black-bellied malkoha (Phướn nhỏ, Phaenicophaeus diardi)
diardi New Latin, formed as the genitive singular of a Latinization of Diard, after Pierre-Médard Diard (1794–1863), French naturalist and explorer.

2/ Green-billed malkoha (Phướn lớn, Phaenicophaeus tristis)
tristis sad, unhappy, melancholy

Surniculus is a small genus of birds in the cuckoo family whose four members are found in tropical Asia and the Philippines. 

1/ Square-tailed drongo-cuckoo (Cu cu đen, Surniculus lugubris)