We establish sufficient criteria for the existence of a limit cycle in the Lienard system ˙x = y − εF(x),˙y = −x, where F(x) is odd. In their simplest form the criteria lead to the result that, for all finite nonzero ε, the amplitude of the limit cycle is less than ρ and 0 a ρ u, where F(a) = 0 and integral from 0 to u of F(x)dx = 0. We take the van der Pol oscillator as a specific example and establish that for all finite, nonzero ε, the amplitude of its limit cycle is less than 2.0672, a value whose precision is limited by the capacity of our symbolic computation software package. We show how the criterion for the upper bound can be extended to establish a bound on the amplitude of a limit cycle in systems where F(x) contains both odd and even components. We also show how the criteria can be used to establish bounds for bifurcation sets.