This book discusses five closely related sets of prime ideals associated to an ideal $I$ in a Noetherian ring: the persistent, asymptotic, quintasymptotic, essential, and quintessential primes of $I$. Since the appearance of the author's last book on this subject, which focused on the first two of these prime ideals, the other three sets were developed and new results were obtained for the first two. Current results are scattered over some three dozen...