Synthetic biology has the potential to positively transform society in many application areas, including medicine. In common with all revolutionary new technologies, synthetic biology can also enable crime. Like cybercrime, that emerged following the advent of the internet, biocrime can have a significant effect on society, but may also impact on peoples' health. For example, the scale of harm caused by the SARS-CoV-2 pandemic illustrates the potential impact of future biocrime and highlights the need for prevention strategies. Systematic evidence quantifying the crime opportunities posed by synthetic biology has to date been very limited. Here, we systematically reviewed forms of crime that could be facilitated by synthetic biology with a view to informing their prevention. A total of 794 articles from four databases were extracted and a three-step screening phase resulted in 15 studies that met our threshold criterion for thematic synthesis. Within those studies, 13 exploits were identified. Of these, 46% were dependent on technologies characteristic of synthetic biology. Eight potential crime types emerged from the studies: bio-discrimination, cyber-biocrime, bio-malware, biohacking, at-home drug manufacturing, illegal gene editing, genetic blackmail, and neuro-hacking. 14 offender types were identified. For the most commonly identified offenders (>3 mentions) 40% were outsider threats. These observations suggest that synthetic biology presents substantial new offending opportunities. Moreover, that more effective engagement, such as ethical hacking, is needed now to prevent a crime harvest from developing in the future. A framework to address the synthetic biology crime landscape is proposed.