An equality between inertial and gravitational masses was established by Galileo Galilei more than 400 years ago and was accepted by Albert Einstein as a key point of his theory of gravitation - General Relativity. The above mentioned equality is called the Equivalence Principle. In this pioneering book, some unusual situations are described, where the Equivalence Principle is theoretically broken, and the possible experiments, where such breakdowns can be observed, are discussed in a brief. It is known that, in standard situations, the Equivalence Principle is extremely well established on Earth and in space in numerous experiments, including experiments during the recent space mission MICROSCOPE. Therefore, this book suggests a real breakthrough in the better understanding of Einstein's gravitational theory and its relation to quantum mechanics, which is a definite step towards the so-called "Theory of Everything". This book is recommended for all readers who are interested in gravitation and General Relativity.