We derive and investigate the microscopic model of the quantum magnet BiCu2PO6 using band structure calculations, magnetic susceptibility and high-field magnetization measurements, as well as Exact Diagonalization (ED) and Density-Matrix Renormalization Group (DMRG) techniques. The resulting quasi-one-dimensional spin model is a two-leg antiferromagnetic ladder with frustrating next-nearest-neighbor couplings along the legs. The individual couplings are estimated from band structure calculations and by fitting the magnetic susceptibility with theoretical predictions, obtained using full diagonalizations. The nearest-neighbor leg coupling J1, the rung coupling J4, and one of the next-nearest-neighbor couplings J2 amount to 120 − 150 K, while the second nextnearest-neighbor coupling is J ′ 2 ≃ J2/2. The spin ladders do not match the structural chains, and although the next-nearest-neighbor interactions J2 and J ′ 2 have very similar superexchange pathways, they differ substantially in magnitude due to a tiny difference in the O-O distances and in the arrangement of non-magnetic PO4 tetrahedra. An extensive ED study of the proposed model provides the low-energy excitation spectrum and shows that the system is in the strong rung coupling regime. The strong frustration by the next-nearest-neighbor couplings leads to a triplon branch with an incommensurate minimum. This is further corroborated by a strong-coupling expansion up to second order in the inter-rung coupling. Based on high-field magnetization measurements, we estimate the spin gap of ∆ ≃ 32 K and suggest the likely presence of antisymmetric DzyaloshinskiiMoriya anisotropy and inter-ladder coupling J3. We also provide a tentative description of the physics of BiCu2PO6 in magnetic field, in the light of the low-energy excitation spectra and numerical calculations based on ED and DMRG. In particular, we raise the possibility for a rich interplay between one-and two-component Luttinger liquid phases and a magnetization plateau at 1/2 of the saturation value.