A model with the Gunfighter advantage can make charges. If it makes a charge, the model can make its initial attacks with its ranged weapons, but these attacks can target only models in its melee range. The charging gunfighter’s first melee or ranged attack after ending its charge movement must target the model it charged. If the charging gunfighter did not fail its charge and cannot make its first melee or ranged attack against the charge target, the charging gunfighter can make its first attack against any eligible target. Remember, your first attack after charging is a charge attack only if you make it with a melee weapon, and a model in melee cannot aim.

Gunfighter does not allow a model to make melee and ranged attacks during the same activation.

2019.12 Errata: The first two sentences were rewritten, in order to clarify the interaction between "in melee range" and "in melee with".
Net result: Gunfighters can now shoot models that are not technically "in melee" (such as knocked down models) but which are in "melee range".