Hungary is a member of the Schengen Agreement.
There are no border controls between countries that have signed and implemented the treaty - the European Union (except Bulgaria, Cyprus, Ireland, Romania and the United Kingdom), Iceland, Liechtenstein, Norway and Switzerland. Likewise, a visa granted for any Schengen member is valid in all other countries that have signed and implemented the treaty. But be careful: not all EU members have signed the Schengen treaty, and not all Schengen members are part of the European Union. This means that there may be spot customs check but no immigration checks (travelling within Schengen but to/from a non-EU country) or you may have to clear immigration but not customs (travelling within the EU but to/from a non-Schengen country).
Airports in Europe are thus divided into "Schengen" and "non-Schengen" sections, which effectively act like "domestic" and "international" sections elsewhere. If you are flying from outside Europe into one Schengen country and continuing to another, you will clear Immigration, but not Customs, at the first country and then continue to your destination where your baggage will have customs checks but there will be no further immigration controls. Travel between a Schengen member and a non-Schengen country will result in the normal border checks. Note that regardless of whether you are travelling within the Schengen area or not, many airlines will still insist on seeing your ID card or passport.
Nationals of EU and EFTA (Iceland, Liechtenstein, Norway, Switzerland) countries only need a valid national identity card or passport for entry - in no case will they need a visa for a stay of any length.
Nationals of non-EU/EFTA countries will generally need a passport for entry to a Schengen country and most will need a visa.
These non-EU/EFTA visa-free visitors may not stay more than 90 days in a 180 day period in the Schengen Area as a whole and, in general, may not work during their stay (although some Schengen countries do allow certain nationalities to work - see below). The counter begins once you enter any country in the Schengen Area and is not reset by leaving a specific Schengen country for another Schengen country, or vice-versa. However, New Zealand citizens may be able to stay for more than 90 days if they only visit particular Schengen countries - see for the New Zealand Government's explanation.
If you are a non-EU/EFTA national (even if you are visa-exempt, unless you are Andorran, Monégasque or San Marinese), make sure that your passport is stamped both when you enter and leave the Schengen Area. Without an entry stamp, you may be treated as an overstayer when you try to leave the Schengen Area; without an exit stamp, you may be denied entry the next time you seek to enter the Schengen Area as you may be deemed to have overstayed on your previous visit. If you cannot obtain a passport stamp, make sure that you retain documents such as boarding passes, transport tickets and ATM slips which may help to convince border inspection staff that you have stayed in the Schengen Area legally.
However, all British Overseas Territories citizens except those solely connected to the Cyprus Sovereign Base Areas are eligible for British citizenship and thereafter unlimited access to the Schengen Area.
Further note that
(*) nationals of Albania, Bosnia and Herzegovina, Macedonia, Montenegro and Serbia need a biometric passport to enjoy visa-free travel,
(**) Serbian nationals with passports issued by the Serbian Coordination Directorate (residents of Kosovo with Serbian passports) do need a visa and
(***) Taiwan nationals need their ID number to be stipulated in their passport to enjoy visa-free travel.
Recognised refugees and stateless persons in possession of a valid travel document issued by the government of any one of the above countries/territories are exempt from obtaining a visa for Hungary (but no other Schengen country, except Germany and, for refugees, Slovakia) for a maximum stay of 90 days in a 180 day period.
Citizens of Antigua and Barbuda are permitted to work in Hungary without the need to obtain a visa for the period of their 90 day visa-free stay. However, this ability to work visa-free does not necessarily extend to other Schengen countries.
Citizens of Croatia can also enter the country by showing their identity card, but may not stay longer than 90 days in a 180-day period or work in Hungary without a work permit.
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