Aim: To examine: (1) relationships between brain structure, and concurrently assessed neurological and behavioural functioning, in infants born preterm at term-equivalent age (TEA; approximately 38–44wks); and (2) whether brain structure–function relationships differ between infants born very (24–29wks) and moderate-late (32–36wks) preterm. Method: A total of 257 infants (91 very preterm, 166 moderate-late preterm; 120 males, 137 females) had structural magnetic resonance imaging (MRI) and neurological and behavioural assessments (Prechtl's general movements assessment, Neonatal Intensive Care Unit Network Neurobehavioral Scale [NNNS] and Hammersmith Neonatal Neurological Examination [HNNE]). Two hundred and sixty-three infants (90 very preterm, 173 moderate-late preterm; 131 males, 132 females) had diffusion MRI and assessments. Associations were investigated between assessment scores and global brain volumes using linear regressions, regional brain volumes using Voxel-Based Morphometry, and white matter microstructure using Tract-Based Spatial Statistics. Results: Suboptimal scores on some assessments were associated with lower fractional anisotropy and/or higher axial, radial, and mean diffusivities in some tracts: NNNS attention and reflexes, and HNNE total score and tone, were associated with the corpus callosum and optic radiation; NNNS quality of movement with the corona radiata; HNNE abnormal signs with several major tracts. Brain structure-function associations generally did not differ between the very and moderate-late preterm groups. Interpretation: White matter microstructural alterations may be associated with suboptimal neurological and behavioural performance in some domains at TEA in infants born preterm. Brain structure–function relationships are similar for infants born very preterm and moderate-late preterm. What this paper adds: Brain volume is not related to neurological/behavioural function in infants born preterm at term. White matter microstructure is related to some neurological/behavioural domains at term. Brain–behaviour relationships are generally similar for infants born very preterm and moderate-late preterm.