Deep-learning (DL) algorithms are becoming the standard for processing ultrasound (US) fetal images. Despite a large number of survey papers already present in this field, 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 145 research papers published after 2017. Each paper is analyzed and commented on from both the methodology and application perspective. We categorized the papers in (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. 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 the actual clinical practice.