Methylammonium lead(II) halides, CH3NH3PbX3 (X = Cl, Br, I), have been investigated by single-crystal X-ray diffraction, 2H and 14N nmr, adiabatic calorimetry, and other methods. The chloride (CL) has transitions at 171.5 and 177.4 K, the bromide (BR) at 148.4, 154.2, and 235.1 K, and the iodide (IO) at 162.7 and 326.6 K. The respective entropies of transition (J K−1 mol−1) are 11.0 and 5.1 for CL; 8.7, 3.4, and 5.3 for BR; and 16.1 and 1.9 for IO. The highest-temperature phase, phase I, of each halide is of the cubic (Pm3m) perovskite type. The cation in CL(I) and BR(I) could not be localized in the electron density maps; the thermal motion of the halogen atom is highly anisotropic. The ln T1(2H) vs. T−1 plots (N-deuterated samples as well as CD3NH3PbCl3) show significant departures from linearity: the temperature variation of T1(2H) in BR(II) and IO(II) can be represented by functions of the type ln T1(H) = k0 − k2T−2, which give adequate analytical representations of T1(2H) and T1(14N) in phase I as well. On cooling, BR(II) and IO(II) exhibit small quadrupole splittings QS(2H), which can be represented to a high degree of correlation by QS(2H) = k(Ttr − T)n, i.e. they appear to exhibit critical behaviour with respect to T. The 14N nmr results indicate that the C—N bond in phase I reorientates in an isotropic potential at a rate approaching that of the freely rotating methylammonium ion. Below phase I this motion takes place in an increasingly anisotropic potential in BR(II) and IO(II) and is essentially arrested in CL(II), BR(III), and IO(III). The temperature dependence of the activation energy Ea for the cation reorientation and other aspects of the non-Arrhenius behaviour are discussed, and the CH3NH3PbX3 perovskites are compared with the corresponding (CH3NH3)2TeX6 halides, utilizing preliminary 2H nmr results on (CH3ND3)2TeBr6. The electrical conductivity, between 0 and 95 °C, of CH3NH3PbI3 increases with temperature and exhibits no discontinuity at Ttr = 326.6 K; the activation energy for the conduction process is estimated as ~0.4 eV. Keywords: crystal structure, MeNH3PbCl3 and MeNH3PbBr3; heat capacity, MeNH3PbX3 (X = Cl, Br, I); methylammonium lead halides; solid-state NMR, 2H and 14N; spin-lattice relaxation, non-Arrhenius behaviour.