We present nine near-infrared (NIR) spectra of supernova (SN) 2005cf at epochs from −10 to +42 d with respect to B-band maximum, complementing the existing excellent data sets available for this prototypical Type Ia SN at other wavelengths. The spectra show a time evolution and spectral features characteristic of normal Type Ia SNe, as illustrated by a comparison with SNe 1999ee, 2002bo and 2003du. The broad-band spectral energy distribution (SED) of SN 2005cf is studied in combined ultraviolet (UV), optical and NIR spectra at five epochs between ∼8 d before and ∼10 d after maximum light. We also present synthetic spectra of the hydrodynamic explosion model W7, which reproduce the key properties of SN 2005cf not only at UV–optical as previously reported, but also at NIR wavelengths. From the radiative-transfer calculations we infer that fluorescence is the driving mechanism that shapes the SED of SNe Ia. In particular, the NIR part of the spectrum is almost devoid of absorption features, and instead dominated by fluorescent emission of both iron-group material and intermediate-mass elements at pre-maximum epochs, and pure iron-group material after maximum light. A single P-Cygni feature of Mg ii at early epochs and a series of relatively unblended Co ii lines at late phases allow us to constrain the regions of the ejecta in which the respective elements are abundant.