Automated Driving Systems (ADS) open up a new domain for the automotive industry and offer new possibilities for future transportation with higher efficiency and comfortable experiences. However, autonomous driving under adverse weather conditions has been the problem that keeps autonomous vehicles (AVs) from going to level 4 or higher autonomy for a long time. This paper assesses the influences and challenges that weather brings to ADS sensors in an analytic and statistical way, and surveys the solutions against inclement weather conditions. State-of-the-art techniques on perception enhancement with regard to each kind of weather are thoroughly reported. External auxiliary solutions, weather conditions coverage in currently available datasets, simulators, and experimental facilities with weather chambers are distinctly sorted out. Additionally, potential future ADS sensors candidates and approaches beyond common senses are provided. By looking into all kinds of major weather problems the autonomous driving field is currently facing, and reviewing both hardware and computer science solutions in recent years, this survey points out the main moving trends of adverse weather problems in autonomous driving, i.e., advanced sensor fusions, more sophisticated networks, and V2X & IoT technologies; and also the limitations brought by emerging 1550 nm LiDARs. In general, this work contributes a holistic overview of the obstacles and directions of ADS development in terms of adverse weather driving conditions.