This article introduces new concepts called the moduli of monotonicity of a real function defined on an interval. They are a one-sided analogue of the well-known modulus of continuity, and are a measure of the extent by which a given function fails to be monotone. It is shown that they naturally arise in the process of approximating a real function by nondecreasing polynomials. Upper and lower bounds on the “degree of approximation” by monotone polynomials are derived in terms of these moduli.