Moscow State Oral Exam I have heard that during the 1960s, prospective students had to take an 'Oral Maths' exam (alongside written maths, physics and Russian literature). 
I having trouble imagining what type of exam this would have been (more problem solving or merely reciting proofs?) and why anybody thought this would be a good idea or a useful indication of potential? 
Was the exam purely oral, as in the candidates were given a problem they had to solve without any written work and on the spot? Or was it of a different format?
I assume it was scrapped but I am not sure when either.
Information on the internet is sparse so I thought I'd ask here, maybe some people even have first hand accounts.
 A: Wow, how so many answers are concentrated on "Jewish problems."
The idea behind the oral exam, which was used in USSR in the places with advanced mathematical education, to understand how good the student will be able to appreciate the real mathematics. The exam was always administered after the written exam. The main point is that it is impossible to cheat at such exam, and it is simply impossible to memorize the material to convince the examiner that you actually understand math.
The structure is as follows: The student picks a random "ticket" with two theoretical questions and one problem. Then he/she is given some time to prepare. And after it the student has to present his solutions to the examiner.
There is a very nice Russian book by Tkachuk (I link a Russian text) about preparation to entrance exams with examples and general discussion. I am not sure though that it was translated into English.
Here is an example of a question: We have a quadrangle, which has all four sides equal. Does it mean that it is a rombus? 
A: The role of oral exams in allowing examiners to accept/dismiss university candidates on the basis of their prejudices has already been mentioned in the comments.  You can read this for a further discussion of this practice.
A: I am working on some research on implementation of oral assessment in mathematics classroom, and I was wondering are there any questions/problems/concepts in mathematics that can be only assessed through Oral Examination (or types of questions where oral assessment would be more sufficient than written)?
A: In the former Soviet Union MIR Publishing helped to develop a number of books to aid in the study of such exams. Most of these books are now public domain. An example of one can be found here. http://mirtitles.org/2013/11/22/problems-in-higher-mathematics-minorsky/
These MIR problem books can be found for various technical subjects and are useful study aids.
