We have developed a generalization of the Zeldovich approximation that is exact in a wide variety of situations, including planar, spherical, and cylindrical symmetries. We have shown that this generalization, which we call the complete Zeldovich approximation (CZA), is exact to second order at an arbitrary point within any field. For Gaussian fields, the third-order error has been obtained and shown to be very small. For statistical purposes, the CZA leads to results exact to the third order.