The assessment of climate variability and change is enriched by novel applications of statistics and machine learning methodologies. This tutorial will be an introduction to some of the common statistical and machine learning problems that arise in climate science. The goal is to give attendees a sense of the intersections between the fields and to help promote future interdisciplinary collaborations. We will introduce you to different climate data sources (e.g., in situ measurements, satellite data, climate model data, etc.) and discuss problems including: characterizing changes in extreme events like heatwaves or extreme precipitation, summarizing high-dimensional spatiotemporal climate data, and using statistical methods to predict climate variability and potentially improve future projections. The focus will be on methodological applications; we will discuss both core methodologies and recent innovations. Prior knowledge of climate science is not assumed and we will emphasize the value of engaging substantively with domain experts.