The recent resurgence of deep learning (DL) has dramatically influenced the medical imaging field. Medical image analysis applications have been at the forefront of DL research efforts applied to multiple diseases and organs, including those of the lungs. The aims of this review are twofold: (i) to briefly overview DL theory as it relates to lung image analysis; (ii) to systematically review the DL research literature relating to the lung image analysis applications of segmentation, reconstruction, registration and synthesis. The review was conducted following the Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines. 479 studies were initially identified from the literature search with 82 studies meeting the eligibility criteria. Segmentation was the most common lung image analysis DL application (65.9% of papers reviewed). DL has shown impressive results when applied to segmentation of the whole lung and other pulmonary structures. DL has also shown great potential for applications in image registration, reconstruction and synthesis. However, the majority of published studies have been limited to structural lung imaging with only 12.9% of reviewed studies employing functional lung imaging modalities, thus highlighting significant opportunities for further research in this field. Although the field of DL in lung image analysis is rapidly expanding, concerns over inconsistent validation and evaluation strategies, intersite generalisability, transparency of methodological detail and interpretability need to be addressed before widespread adoption in clinical lung imaging workflow.