Detection of curvilinear structures has long been of interest due to its wide range of applications. Large amounts of imaging data could be readily used in many fields, but it is practically not possible to analyze them manually. Hence, the need for automated delineation approaches. In the recent years Computer Vision witnessed a paradigm shift from mathematical modelling to data-driven methods based on Machine Learning. This led to improvements in performance and robustness of the detection algorithms. Nonetheless, most Machine Learning methods are general-purpose and they do not exploit the specificity of the delineation problem. In this thesis, we present learning methods suited for this task and we apply them to various kinds of microscopic and natural images, proving the general applicability of the presented solutions. First, we introduce a topology loss - a new training loss term, which captures higher-level features of curvilinear networks such as smoothness, connectivity and continuity. This is in contrast to most Deep Learning segmentation methods that do not take into account the geometry of the resulting prediction. In order to compute the new loss term, we extract topology features of prediction and ground-truth using a pre-trained network, whose filters are activated by structures at different scales and orientations. We show that this approach yields better results in terms of conventional segmentation metrics and overall topology of the resulting delineation. Although segmentation of curvilinear structures provides useful information, it is not always sufficient. In many cases, such as neuroscience and cartography, it is crucial to estimate the network connectivity. In order to find the graph representation of the structure depicted in the image, we propose an approach for joint segmentation and connection classification. Apart from pixel probabilities, this approach also returns the likelihood of a proposed path being a part of the reconstructed network. We show that segmentation and path classification are closely related tasks and can benefit from the synergy. The aforementioned methods rely on Machine Learning, which requires significant amounts of annotated ground-truth data to train models. The labelling process often requires expertise, it is costly and tiresome. To alleviate this problem, we introduce an Active Learning method that significantly decreases the time spent on annotating images. It queries the annotator only about the most informative examples, in this case the hypothetical paths belonging to the structure of interest. Contrary to conventional Active Learning methods, our approach exploits local consistency of linear paths to pick the ones that stand out from their neighborhood. Our final contribution is a method suited for both Active Learning and proofreading the result, which often requires more time than the automated delineation itself. It investigates edges of the delineation graph and determines the ones that are especially significant for the global reconstruction by perturbing their weights. Our Active Learning and proofreading strategies are combined with a new efficient formulation of an optimal subgraph computation and reduce the annotation effort by up to 80%.