This book surveys the current state of the "small" sieve methods developed by Brun, Selberg and later workers.. A self-contained treatment is given to topics that are of central importance in the subject. These include the upper bound method of Selberg, Brun's method, Rosser's sieve as developed by Iwaniec, with a bilinear form of the remainder term, the sieve with weights, and the use of Selberg's ideas in deriving lower-bound sieves. Further developments are introduced with the support of references t. The book is suitable for university graduates making their first acquaintance with the subject, leading them towards the frontiers of modern research and unsolved problems in the subject area.