Number theory could be divided into additive number theory and multiplicative number theory. In this book we raise the so-called additive and multiplicative equation, it is our new creation, and it is the collection and extension of research results by the author in the past decade. They are related with classical number theory problems. The contents consists of 7 chapters, i.e., New Waring Problem, New Fermat Problem, Euler's Conjecture, Representations of Integers as Sums of Squares, Figurate Primes and F-Perfect Numbers, abcd Equation and New Congruent Numbers, Additive and Multiplicative Congruences and Other Topics.