The classical bases in additive number theory are the polygonal numbers, the squares, cubes, and higher powers, and the primes. This book contains many of the great theorems in this subject: Cauchy鈥檚 polygonal number theorem, Linnik鈥檚 theorem on sums of cubes, Hilbert鈥檚 proof of Waring鈥檚 problem, the Hardy-Littlewood asymptotic formula for the number of representations of an integer as the sum of positive kth powers, Shnirel鈥檓an鈥檚 theorem that every integer greater than one is the sum of a bounded number of primes, Vinogradov鈥檚 theorem on sums of three primes, and Chen鈥檚 theorem that every sufficiently large even integer is the sum of a prime and a number that is either prime or the product of two primes. The book is also an introduction to the circle method and sieve methods, which are the principal tools used to study the classical bases. The only prerequisites for the book are undergraduate courses in number theory and analysis. Additive number theory is one of the oldest and richest areas of mathematics. This book is the first comprehensive treatment of the subject in 40 years.