The hadronic environment of the LHC favours the study of exclusive modes, and of these semi-leptonic and radiative B decays will play a leading role in the search for new physics (NP). A prime example is the rare decay B→K∗(→ Kπ)µ+µ−, where the many measurable quantities offer important new tests of the Standard Model and its extensions. We define sets of CP-conserving and CP-violating observables which are studied in terms of the full form factors, calculated in QCD sum rules on the light-cone (LCSR), and QCD factorisation. Those with reduced dependence on hadronic quantities and sensitivity to NP are identified. In the first few years of data–taking at the LHC, the focus will be on quantities which are simple to extract while maximising the available NP sensitivity. Out of three such observables, two are well known to the experimental community. However a third, one of the CP-conserving angular observables, leads to significant additional constraints on parameter space. We then study form factors for rare semi-leptonic and radiative B decays to K(∗) , ρ and φs mesons, combining theoretical and phenomenological constraints from Lattice QCD, LCSR, and dispersive bounds. We pay particular attention to form factor parameterisations which are based on the so-called series expansion, and study the related systematic uncertainties on a quantitative level. Finally we calculate the leading-twist O(αsβ0) corrections to the B → π transition form factor f+ (0) in LCSR, allowing an improved determination of the CKM matrix element |Vub |.