In this chapter we give the fundamentals for a theory of elliptic partial differential equations defined on the boundary of a Lipschitz domain. The key problem will be the Laplace-Beltrami equation − Δ Γ g = f , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/umath17_1.jpg"/> associated to a Dirichlet form ⟨ ∇ Γ g , ∇ Γ h ⟩ Γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math17_1.jpg"/> on a suitably defined Sobolev space H 1(Γ). Apart from the difficulties of defining the surface differential operators, we will be challenged with the property (not easy to prove) that if the surface gradient of a field vanishes, then the field needs to be constant.