The estimation of covariance matrices is a fundamental problem in multivariate analysis and uncertainty quantification. Covariance matrices are an essential modeling tool in climatology, econometrics, model reduction, biostatistics, signal processing, and geostatistics, among other applications. In practice, covariances often must be estimated from samples. While the sample covariance matrix is a consistent estimator, it performs poorly when the relative number of samples is small; improved estimators that impose structure must be considered. Yet standard parametric covariance families can be insufficiently flexible for many applications, and non-parametric approaches may not easily allow certain kinds of prior knowledge to be incorporated. In this thesis, we harness the structure of the manifold of symmetric positive-(semi)definite matrices to build families of covariance matrices out of geodesic curves.