One of the most fascinating challenges in neuroscience is the reconstruction of the connectivity map of the brain. Recent years have seen a rapid expansion in the field of connectomics, whose aim is to trace this map and understand its relationship with neural computation. Many different approaches, ranging from electron and optical microscopy to magnetic resonance imaging, have been proposed to address the connectomics challenge on various spatial scales and in different species. Here, we review the main technological advances in the microscopy techniques applied to connectomics, highlighting the potential and limitations of the different methods. Finally, we briefly discuss the role of connectomics in the Human Brain Project, the Future and Emerging Technologies (FET) Flagship recently approved by the European Commission.