Number theory is a branch of mathematics that is primarily focused on the study of positive integers and their properties. Thus it has often been considered as the purest branch of mathematics in the sense that it was the furthest from any specific application. However, nowadays number theory is considered to be one of the most important branches of mathematics for application in cryptography. This book is based on teaching materials from an undergraduate course Elementary Number Theory and graduate courses Diophantine equations and Diophantine Approximations and Application. The book also provides an insight into subjects that are at the centre of research interest of the author of the book as well as other members of the Croatian number theory research group. This book is primarily intended for undergraduate students who attend courses in number theory and its application. Also, it can be useful for advanced high school students who are preparing for competition in mathematics in which at all levels, from the school level competition to the international competition number theory has a significant role. In addition, it is intended for graduate students and researchers in the fields of number theory, algebra and cryptography.