High-speed optical imaging is an indispensable technology for blur-free observation of fast transient dynamics in virtually all areas including science, industry, defense, energy, and medicine. High temporal resolution is particularly important for microscopy as even a slow event appears to occur "fast" in a small field of view. Unfortunately, the shutter speed and frame rate of conventional cameras based on electronic image sensors are significantly constrained by their electrical operation and limited storage. Over the recent years, several unique and unconventional approaches to high-speed optical imaging have been reported to circumvent these technical challenges and achieve a frame rate and shutter speed far beyond what can be reached with the conventional image sensors. In this article, we review the concepts and principles of such ultrafast optical imaging methods, compare their advantages and disadvantages, and discuss an entirely new class of applications that are possible using them.