Chapter 3 presents a general approach to traces on arbitrary sets. In Sect. 3.1 we first define traces as certain linear continuous functionals. Then we provide a simple but important class of trace functionals over \(\mathcal {L}^\infty (U)\) . They are needed for later use, but initially they also serve for illustration. In Sect. 3.2 we show that \(T_F\) given by \(\displaystyle T_F(f\ ) = \int _{\Omega } f \, d \mathrm {div}F + \int _{\Omega } F\cdot Df\, d \mathcal {L}^{\,n} \) is a trace functional on \(\partial \Omega \) over \(\mathcal {W}^{1,\infty }(U)\) for any \(F\in \mathcal {D}\mathcal {M}^1(U)\) and any Borel set \(\Omega \subset U.\) We also get an analogous result for Sobolev functions in \(\mathcal {W}^{1,1}(U)\) and for BV functions in \(\mathcal {B}\mathcal {V}(U)\) . Section 3.3 is devoted to the representation of such traces by means of measures that are “living” near \(\partial \Omega \) . Here we distinguish three variants of generality called (G), (L), and (C). Theorem 3.14 and several necessary and sufficient conditions for certain special cases are the basis for the subsequent Gauss-Green formulas. Some examples illustrate the spirit behind the three variants.