Since the beginning of this year, the visa-free regime is resumed again for 54 nationalities. Still, all visitors are required to have a negative PRC-Test (even if you're vaccinated).
UPD: February 08, 2022
Restaurants, museums, public venues, airports of Nur-Sultan and Almaty can only be entered if the traveler can prove using the "Ashyq" app that they there are COVID-19 negative or have been in contact with an infected person. This app is available for locals only. All foreigners need a 72hrs PCR test - what literally means you need to undergo PCR testing every 3 days to visit most of venues.

No visa and no permit required for:
Armenia,
Azerbaijan,
Belarus,
Georgia,
Germany,
Hungary,
Kuwait,
Kyrgyzstan,
Netherlands,
Poland,
Russia,
Saudi Arabia,
South Korea,
Tajikistan,
Turkey,
UAE,
Ukraine,
Uzbekistan,
Vietnam
All visa-free nationalities:
Australia
Austria
Bahrain
Belgium
Bulgaria
Canada
Chile
Colombia
Croatia
Cyprus
Czech Republic
Denmark
Finland
France
Germany