Background. The COVID-19 pandemic caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is straining health systems around the world. Although the Chinese government implemented a number of severe restrictions on people's movement in an attempt to contain its local and international spread, the virus had already reached many areas of the world in part due to its potent transmissibility and the fact that a substantial fraction of infected individuals develop little or no symptoms at all. Following its emergence, the virus started to generate sustained transmission in neighboring countries in Asia, Western Europe, Australia, Canada and the United States, and finally in South America and Africa. As the virus continues its global spread, a clear and evidence-based understanding of properties and dynamics of the global transmission network of SARS-CoV-2 is essential to design and put in place efficient and globally coordinated interventions. Methods. We employ molecular surveillance data of SARS-CoV-2 epidemics for inference and comprehensive analysis of its global transmission network before the pandemic declaration. Our goal was to characterize the spatial-temporal transmission pathways that led to the establishment of the pandemic. We exploited a network-based approach specifically tailored to emerging outbreak settings. Specifically, it traces the accumulation of mutations in viral genomic variants via mutation trees, which are then used to infer transmission networks, revealing an up-to-date picture of the spread of SARS-CoV-2 between and within countries and geographic regions. Results and Conclusions. The analysis suggest multiple introductions of SARS-CoV-2 into the majority of world regions by means of heterogeneous transmission pathways. The transmission network is scale-free, with a few genomic variants responsible for the majority of possible transmissions. The network structure is in line with the available temporal information represented by sample collection times and suggest the expected sampling time difference of few days between potential transmission pairs. The inferred network structural properties, transmission clusters and pathways and virus introduction routes emphasize the extent of the global epidemiological linkage and demonstrate the importance of internationally coordinated public health measures.