The Bernoulli Numbers, Euler Numbers, Stirling Numbers, and figurate numbers are originally the result of finite rational sums which are deduced by binomial coefficients and often given by factorials. On the other side, definitions and recurrent formulae with irrational results are existing for these numbers. Especially the Bernoulli Numbers and factorials are of greatest importance in number theory and infinitesimal calculus. The book can show only a small choice of what is thought to be most important.In 9 chapters, after an introduction, Bernoulli and Euler Numbers and Polynomials, infinite series with Bernoulli and Euler Numbers, Bemoulli and Euler Numbers in number theory, a comparison of Bernoulli and Euler Numbers and Polynomials, Stirling Numbers, polygonal, pyramidal, and figurate numbers, factorials and factorial numbers, and different applications are discussed.