This book contains results on the analysis of polynomials and their derivatives. It includes an extensive analysis of orthogonal polynomials as well as of trigonometric sums. Besides the fundamental results which are treated with their proofs, the book also provides an account of the most recent developments concerning external properties of polynomials in various metrics with an extensive analysis of inequalities for polynomials, and their derivatives as well as their zeros. One can also find in this book several new research problems and conjectures with sufficient information concerning the results obtained to date towards the investigation of their solution.