Upcoming gravitational wave detectors are expected to achieve an unprecedented degree of sensitivity. Thus waveforms models determined by numerical relativity will need to be much more accurate. This dissertation presents several techniques to address that need. First, we present a new procedure to extract the complete curvature information of a numerically determined spacetime. The curvature information is expressed by the five Weyl scalars, whose asymptotic form we obtain accurately by extrapolation along approximate null rays. The extraction procedure has been implemented in the Spectral Einstein Code and is straightforward to implement in other numerical relativity codes. The extrapolation procedure is performed in post-processing and made publicly available through the scri python module. Having the asymptotic Weyl scalars along with the asymptotic gravitational-wave strain allows numerical waveforms to be tested against exact general relativity relations and provides an understanding of the error in the underlying calculations. Second, a new Cauchy characteristic extraction (CCE) code was recently developed for computing asymptotic waveforms with greater accuracy than extrapolation. This code should be robust, reliable, and fast enough to replace extrapolation for creating large-scale waveform catalogs. However, a rigorous comparison of the two methods must be performed first to confirm the superiority of CCE. To that end, we present a new waveform catalog that provides asymptotic waveforms for the Weyl scalars and strain, produced by both the new CCE code and our implementation of extraction-extrapolation (Ext). Using this catalog, we present a comparison of the two methods and demonstrate the superiority of CCE. We also discuss the current issues that remain in CCE. Third, in addition to higher accuracy, the CCE procedure is able to better resolve the contributions of gravitational memory in the strain waveform. The gravitational memory effect has been known theoretically for a long time, but up till now no numerical codes have been able to calculate it. Since CCE is much more complicated than our Ext code, we present a simple procedure for correcting the main memory contribution for extrapolated strain waveforms. In addition to this, we present a method for incorporating a smaller contribution, the so-called spin memory, if the Weyl scalar Psi2 has also been extracted. Finally, we use data from our new waveform catalog to compute accurate measurements of the properties of a remnant black hole in numerical relativity. These are the first measurements of the mass, spin, and recoil velocity of the merger remnant using the full set of asymptotic Weyl scalars. We demonstrate a significant improvement in the measurement of the recoil velocity over other methods in the literature. We also demonstrate a remarkable agreement in the measurement of the remnant mass and spin computed from the quasi-local apparent horizon data and from our asymptotic data. By comparing these two independent measurements, we are able to provide a new measure of the error in the remnant properties.