Every positive integer is the sum of four squares. An integral positive definite quadratic form that represents every positive integer is called universal (over the rationals). This notion generalizes to arbitrary totally real fields. It is well-known that that every totally real number field admits a universal quadratic form. For infinite extensions the situation is fundamentally different. Daans, Kala and Man showed that in this case the Northcott property is an obstruction to the existence of such a form. However, Northcott fields are very rare (in a suitable topological sense). We present a necessary condition for the existence of a universal quadratic form in a given number of variables which is new, even in the case of number fields. As an application we show that most totally real fields do not admit a universal quadratic form. This is joint work with Nicolas Daans, Siu Hang Man, Vitezslav Kala, and Pavlo Yatsyna.