Costs
The current exchange rate of the British Pound (GBP) will make perhaps the biggest difference to foreign visitors, but even then York is, by English and global standards, fairly expensive for visitors.
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At the budget end (staying in hostels, rarely eating out, walking and visiting mostly free sights), you might be able to get away with about £60/person/day in high season.
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A more average budget for most visitors (staying at standard hotels or B&Bs, eating out once a day, going to the full range of sights and taking public transport), would be around £150/person/day.
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At the upper end, double that again. With £300/person/day you'll get a pretty high-end experience, with smart hotels, meals at the full range of restaurants, sights and taking taxis around town whenever you feel like it.
Note that travelling solo will probably raise any of these per-person-costs for all except budget travellers.
Visit York Pass
There is now an improved Visit York Pass which may help to cut costs if you intend visiting several of the big attractions.
The York Pass is valid for one, two or three days and includes sightseeing buses and all the main museums, the Minster and Jorvik. The one day adult pass costs around £60 and children pay £35. The adult two and three day passes cost £75 and £90, include local transport and sights around York as well. For up-to-date information visit the website (link below).