Growth of lithium fluoride by molecular beam epitaxy on the (1 1 1) surface of calcium fluoride crystals was studied by TEM and LEED for crystal temperatures from 400 to 773 K and impinging lithium fluoride fluxes from 3×10 11 to 3×10 14 cm −2 s −1. Growth starts, usually, at the 〈1 1 0〉 steps on the (1 1 1) surface of CaF 2. For larger step distances and at later growth stages also growth on the terraces between the steps is found. Preferably, longish, roof-like crystallites are formed, which can be interpreted by growth of LiF(2 0 1 ̄ )[0 1 0] parallel to CaF 2(1 1 1)[ 1 ̄ 0 1]. To a lesser extent square crystallites, i.e. growth with LiF(0 0 1), and, rarely, three-folded pyramidal crystallites, i.e. growth with LiF(1 1 1) parallel to CaF 2(1 1 1), are observed. While the pyramidal crystallites show strict epitaxial orientation with LiF[ 1 ̄ 0 1]‖CaF 2[ 1 ̄ 0 1] and LiF[ 1 ̄ 0 1]‖CaF 2[1 2 ̄ 1], only about 80% of the square crystallites exhibit an epitaxial alignment, where LiF[1 0 0]‖CaF 2[ 1 ̄ 0 1] is preferred to LiF[1 1 0]‖CaF 2[ 1 ̄ 0 1]. The epitaxial relationships are discussed on the basis of theoretically calculated adsorption positions of the lithium fluoride monomer and dimer on the terrace and at the steps of the CaF 2(1 1 1) surface.