Preventing the materialization of climate change is one of the main challenges of our time. The involvement of the financial sector is a fundamental pillar in this task, which has led to the emergence of a new field in the literature, climate finance. In turn, the use of Machine Learning (ML) as a tool to analyze climate finance is on the rise, due to the need to use big data to collect new climate-related information and model complex non-linear relationships. Considering the proliferation of articles in this field, and the potential for the use of ML, we propose a review of the academic literature to assess how ML is enabling climate finance to scale up. The main contribution of this paper is to provide a structure of application domains in a highly fragmented research field, aiming to spur further innovative work from ML experts. To pursue this objective, first we perform a systematic search of three scientific databases to assemble a corpus of relevant studies. Using topic modeling (Latent Dirichlet Allocation) we uncover representative thematic clusters. This allows us to statistically identify seven granular areas where ML is playing a significant role in climate finance literature: natural hazards, biodiversity, agricultural risk, carbon markets, energy economics, ESG factors & investing, and climate data. Second, we perform an analysis highlighting publication trends; and thirdly, we show a breakdown of ML methods applied by research area.