Jet fragmentation transverse momentum (jT) distributions are measured in proton-proton (pp) and proton-lead (p-Pb) collisions at sNN−−−√ = 5.02 TeV with the ALICE experiment at the LHC. Jets are reconstructed with the ALICE tracking detectors and electromagnetic calorimeter using the anti-kT algorithm with resolution parameter R = 0.4 in the pseudorapidity range |η| < 0.25. The jT values are calculated for charged particles inside a fixed cone with a radius R = 0.4 around the reconstructed jet axis. The measured jT distributions are compared with a variety of parton-shower models. Herwig and PYTHIA 8 based models describe the data well for the higher jT region, while they underestimate the lower jT region. The jT distributions are further c...