Climate change has become mankind's main challenge. Greenhouse gas (GHG) emissions from shipping are not totally irresponsible for this representing, roughly, 3% of the global total; an amount equal to that of Germany's total GHG emissions. The Fourth Greenhouse Gas Study 2020 of the International Maritime Organization (IMO) predicts that the share of GHG emissions from shipping will increase further, as international trade recovers and continues to grow, alongside with the economic development of India, China, and Africa. China and the European Union have proposed to include shipping in their carbon emissions trading systems (ETS). As a result, the study of the relationship between the carbon finance market and the shipping industry, attempted here for the first time, is both important and timely, both for policymakers and shipowners. We use wavelet analysis and the spillover index methods to explore the dynamic dependence and information spillovers between the carbon finance market and shipping. We discover a long-term dependence and information linkages between the two markets, with the carbon finance market being the dominant one. Major events, such as the 2009 global financial crisis; Brexit in 2016; the 2018 China-US trade frictions; and COVID-19 are shown to strengthen the dependence of carbon finance and shipping. We find that the dependence is strongest between the EU carbon finance market and dry bulk shipping, while the link is weaker in the case of tanker shipping. Nonetheless, carbon finance and tanker shipping showed a relatively stronger dependence when OPEC refused to cut production in 2014, and when the China-US trade disputes led to the collapse of oil prices after 2018. We show that information spillovers between carbon finance and shipping are bidirectional and asymmetric, with the carbon finance market being the principal transmitter of information. Our results and their interpretation provide guidance to governments on whether (and how) to include shipping in emissions trading schemes, supporting at the same time the environmental sustainability decisions of shipping companies.