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Modeling the dynamics of complex multibody systems, such as those representing the motion of animals, can be accomplished through well-established geometric methods. In these formulations, motions take values in certain types of smooth manifolds which are coordinate-free and intrinsic. However, the dimension of the full configuration manifold can be large. The first study in this dissertation aims to build low-dimensional models models from motion capture data. This study also expands on the so-called learning problem from statistical learning theory over Euclidean spaces to estimating functions over manifolds. Experimental results are presented for estimating reptilian motion using motion capture data. The second study in this dissertation utilizes reproducing kernel Hilbert space (RKHS) formulations and Koopman theory, to achieve some of the advantages of learning theory for IID discrete systems to estimates generated over dynamical systems. Specifically, rates of convergence are determined for estimates generated via extended dynamic mode decomposition (EDMD) by relating them to estimates generated by distribution-free learning theory. Some analytical examples illustrate the qualitative behavior of the estimates. Additionally, a examination of the numerical stability of the estimates is also provided in this study. The approximation methods are then implemented to estimate forward kinematics using motion capture data of a human running along a treadmill. The final study of this dissertation contains an examination of the continuous time regression problem over subsets and manifolds. Rates of convergence are determined using a new notion of Persistency of Excitation over flows of manifolds. For practical considerations, two approximation methods of the exact solution to the continuous regression problem are introduced. Characteristics of these approximation methods are analyzed using numerical simulations. Implementations of the approximation schemes are also performed on experimentally collected motion capture data. Doctor of Philosophy Modeling the dynamics of complex multibody systems, such as those representing the motion of animals, can be accomplished through well-established geometric methods. However, many real-world systems, including those representing animal motion, are difficult to model from first principles. Machine learning, on the other hand, has proven to be extremely powerful in its ability to leverage "big data" to generate estimates from typically independent and identically distributed (IID) data. This dissertation expands on the so-called learning problem from statistical learning theory over Euclidean spaces to those over manifolds. This dissertation consists of three studies, the first of which aims to build low-dimensional models models from motion capture data. Using the distribution-free learning theory, estimates discussed in this dissertation minimize a proxy of the expected error, which cannot be calculated in closed form. This dissertation also includes a study into approximations of the so-called Koopman operator. This study determined that the rate of convergence of the estimate to the true operator depends on the reduced dimensionality of the embedded submanifold in the high-dimensional ambient input space. While most of the current work on machine learning focuses on cases where the samples used for learning or regression are generated from an IID, stochastic, discrete measurement process, this dissertation also contains a study of the regression problem in continuous time over subsets and manifolds. Additionally, two approximation methods of the exact solution to the continuous regression problem are introduced. Each of the aforementioned studies also includes several analytical results to illustrate the qualitative behavior of the approximations and, in each study, implementations of the estimation schemes are performed on experimentally collected motion capture data.