European Medical Center is the highest rated private hospital in Moscow. They offer 24/7 healthcare and outstanding service in English. They have several locations in the city, including a special kids clinic and birth center. They are very flexible when it comes to scheduling appointments and you can choose your preferred time, location and doctor. They have high tech equipment and very modern facilities.
They are very thorough, which is good to a certain extent. Do be aware that they sometimes seem to apply extra checks and treatment for commercial reasons, especially when they now you're insurance will cover it. So we advice you to use your common sense and sometimes ask for a second opinion in your own country, because this might save some unnecessary CT or MRI scans ;)