An increasing amount of attention has been geared towards understanding the privacy risks that arise from sharing genomic data of human origin. Most of these efforts have focused on issues in the context of genomic sequence data, but the popularity of techniques for collecting other types of genome-related data has prompted researchers to investigate privacy concerns in a broader genomic context. In this review, we give an overview of different types of genome-associated data, their individual ways of revealing sensitive information, the motivation to share them as well as established and upcoming methods to minimize information leakage. We further discuss the concise threats that are being posed, who is at risk, and how the risk level compares to potential benefits, all while addressing the topic in the context of modern technology, methodology, and information sharing culture. Additionally, we will discuss the current legal situation regarding the sharing of genomic data in a selection of countries, evaluating the scope of their applicability as well as their limitations. We will finalize this review by evaluating the development that is required in the scientific field in the near future in order to improve and develop privacy-preserving data sharing techniques for the genomic context.