Monolayer molybdenum disulfide (MoS2) has attracted tremendous attention due to its promising applications in high-performance field-effect transistors, phototransistors, spintronic devices and nonlinear optics. The enhanced photoluminescence effect in monolayer MoS2 was discovered and, as a strong tool, was employed for strain and defect analysis in MoS2. Recently, large-size monolayer MoS2 has been produced by chemical vapour deposition, but has not yet been fully explored. Here we systematically characterize chemical vapour deposition-grown MoS2 by photoluminescence spectroscopy and mapping and demonstrate non-uniform strain in single-crystalline monolayer MoS2 and strain-induced bandgap engineering. We also evaluate the effective strain transferred from polymer substrates to MoS2 by three-dimensional finite element analysis. Furthermore, our work demonstrates that photoluminescence mapping can be used as a non-contact approach for quick identification of grain boundaries in MoS2.