The ICT sector, responsible for 2% of global carbon emissions, is under scrutiny calling for methodologies and tools to design and develop software in an environmentally sustainable-by-design manner. However, the software engineering solutions for designing and developing carbon-efficient software are currently scattered over multiple different pieces of literature, which makes it difficult to consult the body of knowledge on the topic. In this article, we precisely conduct a systematic literature review on state-of-the-art proposals for designing and developing carbon-efficient software. We identify and analyse 65 primary studies by classifying them through a taxonomy aimed at answering the 5W1H questions of carbon-efficient software design and development. We first provide a reasoned overview and discussion of the existing guidelines, reference models, measurement solutions and techniques for measuring, reducing, or minimising the carbon footprint of software. Ultimately, we identify open challenges and research gaps, offering insights for future work in this field.