We consider total well-founded orderings on monadic terms satisfying the replacement and full invariance properties. We show that any such ordering on monadic terms in one variable and two unary function symbols must have order type ω, ω² or ω^ω. We show that a familiar construction gives rise to continuum many such orderings of order type ω. We construct a new family of such orderings of order type ω², and show that there are continuum many of these. We show that there are only four such orderings of order type ω^ω, the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings on Nⁿ which are preserved under vector addition. We show that any such ordering must have order type ω^k for some 1 ≤ k ≤ n. We show that if <latex>$k < n$</latex> there are continuum many such orderings, and if k = n there are only n!, the n! lexicographic orderings.