Abstract Delineation of curvilinear structures is an important problem in Computer Vision with multiple practical applications. With the advent of Deep Learning, many current approaches on automatic delineation have focused on finding more powerful deep architectures, but have continued using the habitual pixel-wise losses such as binary crossentropy. In this paper we claim that pixel-wise losses alone are unsuitable for this problem because of their inability to reflect the topological impact of mistakes in the final prediction. We propose a new loss term that is aware of the higherorder topological features of linear structures. We also exploit a refinement pipeline that iteratively applies the same model over the previous delineation to refine the predictions at each step, while keeping the number of parameters and the complexity of the model constant. When combined with the standard pixel-wise loss, both our new loss term and an iterative refinement boost the quality of the predicted delineations, in some cases almost doubling the accuracy as compared to the same classifier trained with the binary cross-entropy alone. We show that our approach outperforms state-of-the-art methods on a wide range of data, from microscopy to aerial images