Climate change triggers a wide range of hydrometeorological, glaciological and geophysical processes that span across vast spatiotemporal scales. With the advances in technology and analytics, a multitude of remote sensing, geodetic and in situ instruments have been developed to effectively monitor and help comprehend the Earth's system including its climate variability and the recent anomalies associated with global warming. A huge volume of data is generated by recording these observations, resulting in the need for novel methods to handle and interpret such Big Datasets. Managing this enormous amount of data extends beyond current computer storage considerations; it also encompasses the complexities of processing, modeling, and analysing. Big Datasets present unique characteristics that set them apart from smaller datasets, thereby posing challenges to traditional approaches. Moreover, computational time plays a crucial role, especially in the context of geohazard warning and response systems which necessitate low latency requirements. In this review, we delve into the monitoring and analysis of various climate change-related phenomena, including, but not limited to, droughts, floods, cyclones-induced storm surges, urban heat islands, ice mass balance, sea-level rise, and the modelling of the influence of solar variability on the Earth’s climate. By examining these phenomena, we explore some of the current and future trends in Big Data, aiming to encourage and speed-up the development of such techniques and promoting their benefits to timely monitor and towards achieving climate sustainability, thereby addressing its threat to humanity.