My father is an Italian national and I have a valid Carta di soggiorno di familiare di un cittadino dell'Unione of Italy. I am currently residing in Turkey and I would be travelling to Hungary next month (without my father).
Will I be needing visa?
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If your card is indeed valid, you do not need a visa to travel to Hungary. From the Schengen Borders Code:
‘residence permit’ means: (a) ... and residence cards issued in accordance with Directive 2004/38/EC;
The residence card you hold, having been issued in accordance with Directive 2004/38/EC, is therefore a "residence permit" for the purpose of the Schengen Borders Code.
For intended stays on the territory of the Member States of a duration of no more than 90 days in any 180-day period, which entails considering the 180-day period preceding each day of stay, the entry conditions for third-country nationals shall be the following:
(a) [passport requirements omitted as irrelevant to the question, but be aware that you'll need your passport.]
(b) they are in possession of a valid visa, if required pursuant to Council Regulation (EC) No 539/2001 ( 5 ), except where they hold a valid residence permit or a valid long-stay visa;
This answers your question, in combination with the above definition of "residence permit": Your residence card exempts you from any requirement to hold a visa.
The remaining items in the list specify that you will need to "justify the purpose and conditions of the intended stay" as well as that you have "sufficient means of subsistence", that you haven't been listed in the Schengen Information System "for the purposes of refusing entry," and that you "are not considered to be a threat to public policy, internal security, public health or the international relations of any of the Member States."
You should be careful, because if you live outside Italy for a certain period of time, you may lose your status as a resident even if the card hasn't expired. This is controlled by directive 2004/38/EC.
In particular, according to Article 11(2):
The validity of the residence card shall not be affected by temporary absences not exceeding six months a year, or by absences of a longer duration for compulsory military service or by one absence of a maximum of 12 consecutive months for important reasons such as pregnancy and childbirth, serious illness, study or vocational training, or a posting in another Member State or a third country.
This means that if you are out of the country for over twelve months for one of the enumerated reasons, or for over six months for any other reason, you run the risk of losing your residence. If you've acquired permanent residence, though, you can remain outside Italy for up to two years. This is controlled by Article 16(4):
Once acquired, the right of permanent residence shall be lost only through absence from the host Member State for a period exceeding two consecutive years.