The aim of this book is to study some algebraic concepts such as, almost prime ideals, generalized left and right primary rings and give some generalizations of them. Also Some new algebraic structures are introduced such as, almost primary ideals, generalized k-primary ideals (g.k.p. ideals), generalized k-primary rings (g.k.p. rings), principal generalized k-primary ideals (p.g.k.p. ideals), principal generalized k-primary rings (p.g.k.p. rings), completely generalized k-primary ideals (c.g.k.p. ideals), completely generalized k-primary rings (c.g.k.p. rings). Several properties of these new algebraic structures are proved and many characterizations of them are given. Also, some conditions are obtained which make two types or more of these new concepts equivalent. Finally, the relationships of these new algebraic structures with prime rings, semiprime rings, reduced rings and strongly regular rings are studied and some results are proved.