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:
All visa-free nationalities: