This survey paper covers the breadth and depth of time-series and spatiotemporal causality methods, and their applications in Earth Science. More specifically, the paper presents an overview of causal discovery and causal inference, explains the underlying causal assumptions, and enlists evaluation techniques and key terminologies of the domain area. The paper elicits the various state-of-the-art methods introduced for time-series and spatiotemporal causal analysis along with their strengths and limitations. The paper further describes the existing applications of several methods for answering specific Earth Science questions such as extreme weather events, sea level rise, teleconnections etc. This survey paper can serve as a primer for Data Science researchers interested in data-driven causal study as we share a list of resources, such as Earth Science datasets (synthetic, simulated and observational data) and open source tools for causal analysis. It will equally benefit the Earth Science community interested in taking an AI-driven approach to study the causality of different dynamic and thermodynamic processes as we present the open challenges and opportunities in performing causality-based Earth Science study.