Machine Learning and Deep Learning methods are widely adopted across financial domains to support trading activities, mobile banking, payments, and making customer credit decisions. These methods also play a vital role in combating financial crime, fraud, and cyberattacks. Financial crime is increasingly being committed over cyberspace, and cybercriminals are using a combination of hacking and social engineering techniques which are bypassing current financial and corporate institution security. With this comes a new umbrella term to capture the evolving landscape which is financial cybercrime. It is a combination of financial crime, hacking, and social engineering committed over cyberspace for the sole purpose of illegal economic gain. Identifying financial cybercrime-related activities is a hard problem, for example, a highly restrictive algorithm may block all suspicious activity obstructing genuine customer business. Navigating and identifying legitimate illicit transactions is not the only issue faced by financial institutions, there is a growing demand of transparency, fairness, and privacy from customers and regulators, which imposes unique constraints on the application of artificial intelligence methods to detect fraud-related activities. Traditionally, rule based systems and shallow anomaly detection methods have been applied to detect financial crime and fraud, but recent developments have seen graph based techniques and neural network models being used to tackle financial cybercrime. There is still a lack of a holistic understanding of the financial cybercrime ecosystem, relevant methods, and their drawbacks and new emerging open problems in this domain in spite of their popularity. In this survey, we aim to bridge the gap by studying the financial cybercrime ecosystem based on four axes: (a) different fraud methods adopted by criminals; (b) relevant systems, algorithms, drawbacks, constraints, and metrics used to combat each fraud type; (c) the relevant personas and stakeholders involved; (d) open and emerging problems in the financial cybercrime domain.