Polyatomic molecules have rich structural features that make them uniquely suited to applications in quantum information science1–3, quantum simulation4–6, ultracold chemistry7 and searches for physics beyond the standard model8–10. However, a key challenge is fully controlling both the internal quantum state and the motional degrees of freedom of the molecules. Here we demonstrate the creation of an optical tweezer array of individual polyatomic molecules, CaOH, with quantum control of their internal quantum state. The complex quantum structure of CaOH results in a non-trivial dependence of the molecules’ behaviour on the tweezer light wavelength. We control this interaction and directly and non-destructively image individual molecules in the tweezer array with a fidelity greater than 90%. The molecules are manipulated at the single internal quantum state level, thus demonstrating coherent state control in a tweezer array. The platform demonstrated here will enable a variety of experiments using individual polyatomic molecules with arbitrary spatial arrangement. An optical tweezer array of individual polyatomic molecules is created, revealing the obvious state control in the tweezer array and enabling further research on polyatomic molecules with diverse spatial arrangements.