With the proliferation of smart devices and revolutions in communications, electrical distribution systems are gradually shifting from passive, manually-operated and inflexible ones, to a massively interconnected cyber-physical smart grid to address the energy challenges of the future. However, the integration of several cutting-edge technologies has introduced several security and privacy vulnerabilities due to the large-scale complexity and resource limitations of deployments. Recent research trends have shown that False Data Injection (FDI) attacks are becoming one of the most malicious cyber threats within the entire smart grid paradigm. Therefore, this paper presents a comprehensive survey of the recent advances in FDI attacks within active distribution systems and proposes a taxonomy to classify the FDI threats with respect to smart grid targets. The related studies are contrasted and summarized in terms of the attack methodologies and implications on the electrical power distribution networks. Finally, we identify some research gaps and recommend a number of future research directions to guide and motivate prospective researchers.