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
Greece
Hungary
Iceland
Indonesia
Ireland
Israel
Italy
Japan
Korea
Kuwait
Latvia
Liechtenstein
Lithuania
Luxembourg
Malaysia
Malta
Mexico
Monaco
Netherlands
New Zealand
Norway
Oman
Philippines
Poland
Portugal
Qatar
Romania
Saudi Arabia
Singapore
Slovakia
Slovenia
Spain
Sweden
Switzerland
Thailand
Turkey
UAE
UK
USA
Vatican City
Vietnam