Non-Invasive foetal electrocardiography (NI-FECG) represents an alternative foetal monitoring technique to traditional Doppler ultrasound approaches, that is non-invasive and has the potential to provide additional clinical information. However, despite the significant advances in the field of adult ECG signal processing over the past decades, the analysis of NI-FECG remains challenging and largely unexplored. This is mainly due to the relatively low signal-to-noise ratio of the FECG compared to the maternal ECG, which overlaps in both time and frequency. This article is intended to be used by researchers as a practical guide to NI-FECG signal processing, in the context of the above issues. It reviews recent advances in NI-FECG research including: publicly available databases, NI-FECG extraction techniques for foetal heart rate evaluation and morphological analysis, NI-FECG simulators and the methodology and statistics for assessing the performance of the extraction algorithms. Reference to the most recent work is given, recent findings are highlighted in the form of intermediate summaries, references to open source code and publicly available databases are provided and promising directions for future research are motivated. In particular we emphasise the need and specifications for building a new open reference database of NI-FECG signals, and the need for new algorithms to be benchmarked on the same database, employing the same evaluation statistics. Finally we motivate the need for research in NI-FECG to address morphological analysis, since this represent one of the most promising avenues for this foetal monitoring modality.