In this talk we will discuss the problem of determining the existence of quadratic points on the intersection of two quadrics in five projective variables. The answer to this question heavily depends on the arithmetic and complexity of the base field. In particular, we will explain the differences between the cases of finite fields, p-adic fields, and number fields, and we will conclude with a result over global function fields obtained in joint work with Katerina Santicola, Harry Shaw and Haowen Zhang.