Many, see text
Acanthis Linaria Chloris (but see article text)
The genus Carduelis is a large group of birds in the finch family Fringillidae. It includes the greenfinches, redpolls, goldfinches, linnets, the twite, and the non-African siskins. No species of this group ranges far into Africa (where they are replaced by the related genus Serinus), and the centers of evolution were probably Eurasia and North America, with a secondary radiation in the Neotropics.
The interrelationship of these species is complex and contentious. It is fairly certain that the crossbills are actually derived from proto-redpoll ancestors quite recently, and it was suggested that they should be placed in the same genus, for which the name Loxia would then have priority. On the other hand, the greenfinches (which are apparently the most distinct group) and the redpolls have themselves been separated in distinct genera which might be the best way to express both the actual evolutionary relationships and the evolutionarily significant distinctiveness of the crossbills. The molecular data indicates that the major lineages split in the Late Miocene (Tortonian, roughly 9-7 mya), but it is unable to suggest any one robust arrangement either of the major groups among each other, among the lineages of Carduelis sensu stricto, or indeed among two separate Serinus lineages (Ryan et al, 2004). As only the mitochondrial cytochrome b sequence has hitherto been studied (Arnaiz-Villena et al., 1998), more data is clearly necessary.
Here, the species of Carduelis sensu lato are listed according to current knowledge. The genus Carduelis sensu stricto could conceivably be split further, and in this case only the European Goldfinch and the Citril and Corsican Finch (newly placed in his genus) would remain in Carduelis.

Greenfinches
(Sub)Genus Acanthis

Arctic Redpoll, or Hoary Redpoll, Carduelis hornemanni
Common Redpoll, or Mealy Redpoll, Carduelis flammea
Lesser Redpoll, Carduelis cabaret Redpolls
(Sub)Genus Loxia

3 - 5+ species Crossbills
Carduelis group
Linaria group - linnets and Twite
Spinus group - American goldfinches and siskins
American goldfinches
Northern siskins
Neotropical siskins

European Goldfinch, Carduelis carduelis
Citril Finch, Serinus citrinella
Corsican Finch, Serinus corsicana
Eurasian Linnet, Carduelis cannabina
Warsangli Linnet, Carduelis johannis
Yemen Linnet, Carduelis yemenensis
Twite, Carduelis flavirostris
American Goldfinch, Carduelis tristis
Lesser Goldfinch, Carduelis psaltria
Lawrence's Goldfinch, Carduelis lawrencei
Eurasian Siskin, or Spruce Siskin, Carduelis spinus
Pine Siskin, Carduelis pinus
Andean Siskin, Carduelis spinescens
Antillean Siskin, Carduelis dominicensis
Black Siskin, Carduelis atrata
Black-capped Siskin, Carduelis atriceps
Black-chinned Siskin, Carduelis barbata
Black-headed Siskin, Carduelis notata
Hooded Siskin, Carduelis magellanica
Olivaceous Siskin, Carduelis olivacea
Red Siskin, Carduelis cucullata
Saffron Siskin, Carduelis siemiradzkii
Thick-billed Siskin, Carduelis crassirostris
Yellow-bellied Siskin, Carduelis xanthogastra
Yellow-faced Siskin, Carduelis yarrellii
Yellow-rumped Siskin, Carduelis uropygialis