A norm form is a form F(X" ... ,Xn ) with rational coefficients which factors into linear forms over C but is irreducible or a power of an irreducible form over Q. It is known that a nondegenerate norm form equation F(x" .... xn) = m has only finitely many. solutions (XI, .... X n) E zn. We derive explicit bounds for the number of solutions. When F has coefficients in Z. these bounds depend only on n, m and the degree of F, but are independent of the size of the coefficients of F .