If \(f\in L^ 2({\mathbb{R}})\) then, for \((u,v)\in {\mathbb{R}}^ 2,\) \(A(f)(u,v)=\)defines the radar ambiguity function of f, in \(L^ 2({\mathbb{R}}^ 2)\). The paper studies the properties of ambiguity functions. It is shown that the set of all ambiguity functions is closed in \(L^ 2({\mathbb{R}}^ 2)\) and that if f,g\(\in L^ 2({\mathbb{R}})\), then \(A(f)+A(g)\) can be an ambiguity function if and only if \(f=c g\) where c is a constant. Also, whenever f in \(L^ 2({\mathbb{R}})\) generates an \(L^ 2\)-basis, it is proved that the ambiguity functions are described in terms of A(f), thus enabling a description of all ambiguity functions in terms of well-known functions.