Graph Theory is an important area of contemporary mathematics with many applications in computer science, genetics, chemistry, engineering, industry, business and in social sciences. It is a young science invented and developing for solving challenging problems of 'computerised' society for which traditional areas of mathematics such as algebra or calculus are powerless. This book is for math and computer science majors, for students and representatives of many other disciplines (like bioinformatics, for example) taking the courses in graph theory, discrete mathematics, data structures, algorithms. It is also for anyone who wants to understand the basics of graph theory, or just is curious. No previous knowledge in graph theory or any other significant mathematics is required. The very basic facts from set theory, proof techniques and algorithms are sufficient to understand it; but even those are explained in the text. The book discusses the key concepts of graph theory with emphasis on trees, bipartite graphs, cycles, chordal graphs, planar graphs and graph colouring. The reader is conducted from the simplest examples, definitions and concepts, step by step, towards an understanding of a few most fundamental facts in the field to show an interaction between the sections and chapters for the sake of integrity; clearly expose the essence and core of graph theory. The book may be used on undergraduate level for one semester introductory course. It includes many examples, figures and algorithms; each section ends with a set of exercises and a set of computer projects. The answers and hints to selected exercises are provided at the end of the book. The material has been tested in class during more than 20-years of teaching experience of the author.