We define four types of invexity for Lipschitz vector-valued mappings fromRptoRqthat generalize previous definitions of invexity in the differentiable setting. After establishing relationships between the various definitions, we show the importance of the concept of nonsmooth invexity in the field of optimization. In particular, we obtain conditions sufficient for optimality in unconstrained and cone-constrained nondifferentiable programming that are weaker than previous conditions presented in the literature; we also obtain weak and strong duality results.