Quantifier elimination is a strong and useful tool in classical (commutative) real algebra and geometry. Non-commutative real algebra and geometry is a recently emerging area of research, with many interesting applications in pure and applied mathematics. Quantifier elimination would be a highly desirable tool here as well. There are some results in that direction, mostly negative. We will give a survey on this question, presenting some new results, among which is also a first positive one.

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