Fetal electrocardiography (fECG) is a promising alternative to cardiotocography continuous fetal monitoring. Robust extraction of the fetal signal from the abdominal mixture of maternal and fetal electrocardiograms presents the greatest challenge to effective fECG monitoring. This is mainly due to the low amplitude of the fetal versus maternal electrocardiogram and to the non-stationarity of the recorded signals. In this review, we highlight key developments in advanced signal processing algorithms for non-invasive fECG extraction and the available open access resources (databases and source code). In particular, we highlight the advantages and limitations of these algorithms as well as key parameters that must be set to ensure their optimal performance. Improving or combining the current or developing new advanced signal processing methods may enable morphological analysis of the fetal electrocardiogram, which today is only possible using the invasive scalp electrocardiography method.