Dynamic Time Warping (DTW) is a distance measure that compares two time series after optimally aligning them. DTW is being used for decades in thousands of academic and industrial projects despite the very expensive computational complexity, O(n2). These applications include data mining, image processing, signal processing, robotics and computer graphics among many others. In spite of all this research effort, there are many myths and misunderstanding about DTW in the literature, for example "it is too slow to be useful" or "the warping window size does not matter much." In this tutorial, we correct these misunderstandings and we summarize the research efforts in optimizing both the efficiency and effectiveness of both the basic DTW algorithm, and of the higher-level algorithms that exploit DTW such as similarity search, clustering and classification. We will discuss variants of DTW such as constrained DTW, multidimensional DTW and asynchronous DTW, and optimization techniques such as lower bounding, early abandoning, run-length encoding, bounded approximation and hardware optimization. We will discuss a multitude of application areas including physiological monitoring, social media mining, activity recognition and animal sound processing. The optimization techniques are generalizable to other domains on various data types and problems.