Richard Dedekind published in 1888 a paper entitled "Was sind und was sollen die Zahlen?", variously translated as "What are numbers and what should they be?". In analogy to this classic, I shall investigate in this talk what spinors (or, in full term, spinor fields) are, what distinguishes them from functions, how they appear naturally in complex analysis and theoretical physics, and, finally, why they are an object of intrinsic mathematical interest. Doing so, I will give a gentle introduction to spin geometry and Dirac operators for the non-experts, and I will provide an overview of typical problems and interesting links to other areas.