We study the properties of Carnot–Carathéodory spaces attached to a strictly pseudoconvex CR manifold M, in a neighborhood of each point $$x \in M$$ x∈M , versus the pseudohermitian geometry of M arising from a fixed positively oriented contact form $$\theta $$ θ on M. The weak Dirichlet problem for the sublaplacian $$\Delta _b$$ Δb on $$(M, \theta )$$ (M,θ) is solved on domains $$\Omega \subset M$$ Ω⊂M supporting the Poincaré inequality. The solution to Neumann problem for the sublaplacian $$\Delta _b$$ Δb on a $$C^{1,1}$$ C1,1 connected $$(\epsilon , \delta )$$ (ϵ,δ) -domain $$\Omega \subset {{\mathbb {G}}}$$ Ω⊂G in a Carnot group (due to Danielli et al. in: Memoirs of American Mathematical Society 2006) is revisited for domains in a CR manifold. As an application we prove discreetness of the Dirichlet and Neumann spectra of $$\Delta _b$$ Δb on $$\Omega \subset M$$ Ω⊂M in a Carnot–Carthéodory complete pseudohermitian manifold $$(M, \theta )$$ (M,θ) .