This two-part tutorial will introduce the framework of conformal prediction, and will provide an overview of both theoretical foundations and practical methodologies in this field. In the first part of the tutorial, we will cover methods including holdout set methods, full conformal prediction, cross-validation based methods, calibration procedures, and more, with emphasis on how these methods can be adapted to a range of settings to achieve robust uncertainty quantification without compromising on accuracy. In the second part, we will cover some recent extensions that allow the methodology to be applied in broader settings, such as weighted conformal prediction, localized methods, online conformal prediction, and outlier detection.