Influencer marketing has become a crucial feature of digital marketing strategies. Despite its rapid growth and algorithmic relevance, the field of computational studies in influencer marketing remains fragmented, especially with limited systematic reviews covering the computational methodologies employed. This makes overarching scientific measurements in the influencer economy very scarce, to the detriment of interested stakeholders outside of platforms themselves, such as regulators, but also researchers from other fields. This paper aims to provide an overview of the state of the art of computational studies in influencer marketing by conducting a systematic literature review (SLR) based on the PRISMA model. The paper analyses 69 studies to identify key research themes, methodologies, and future directions in this research field. The review identifies four major research themes: Influencer identification and characterisation, Advertising strategies and engagement, Sponsored content analysis and discovery, and Fairness. Methodologically, the studies are categorised into machine learning-based techniques (e.g., classification, clustering) and non-machine-learning-based techniques (e.g., statistical analysis, network analysis). Key findings reveal a strong focus on optimising commercial outcomes, with limited attention to regulatory compliance and ethical considerations. The review highlights the need for more nuanced computational research that incorporates contextual factors such as language, platform, and industry type, as well as improved model explainability and dataset reproducibility. The paper concludes by proposing a multidisciplinary research agenda that emphasises the need for further links to regulation and compliance technology, finer granularity in analysis, and the development of standardised datasets.