Broadband terahertz time-domain spectroscopy (THz-TDS) has been shown to be a valuable technique for the detection and analysis of explosives. In this paper, we present recent work on the use of this technique to analyse two pure explosives, 1,3,5-trinitroperhydro-1,3,5-triazine (RDX) and 1,3-dinitrato-2,2-bis(nitratomethyl)propane (PETN) and three plastic explosives (Semtex, SX2 and Metabel). For each explosive, a clear and unique spectral absorption fingerprint is found, which can be attributed to oscillations of the molecular lattice. Temperature dependent terahertz absorption measurements of RDX show that the observed vibrational modes are anharmonic in nature. Terahertz spectra are compared with Raman spectra, and the similarities and differences between the spectra produced by the two techniques are discussed. A molecular mechanics calculation approach is used to calculate vibrational intensities and frequencies, and these results are compared with experimental results at both room temperature and 4 K.