Deep-learning (DL) algorithms are becoming the standard for processing ultrasound (US) fetal images. A number of survey papers in the field is today available, but most of them are focusing on a broader area of medical-image analysis or not covering all fetal US DL applications. This paper surveys the most recent work in the field, with a total of 153 research papers published after 2017. Papers are analyzed and commented from both the methodology and the application perspective. We categorized the papers into (i) fetal standard-plane detection, (ii) anatomical structure analysis and (iii) biometry parameter estimation. For each category, main limitations and open issues are presented. Summary tables are included to facilitate the comparison among the different approaches. In addition, emerging applications are also outlined. Publicly-available datasets and performance metrics commonly used to assess algorithm performance are summarized, too. This paper ends with a critical summary of the current state of the art on DL algorithms for fetal US image analysis and a discussion on current challenges that have to be tackled by researchers working in the field to translate the research methodology into actual clinical practice.