This thesis is concerned with operators on certain vector-valued function spaces. Namely, Bergman spaces of \mathbb{C}^n$-valued functions and L^2(\mathbb{R},\mathbb{C}^n,V)$, where $V$ is a matrix weight. We will study products of Toeplitz operators on the vector Bergman space $L^2_a(\mathbb{C}^n)$. We also study various operators, including the dyadic shift and the Hilbert transform, between $L^2(\mathbb{R},\mathbb{C}^n,V)$ and $L^2(\mathbb{R},\mathbb{C}^n,U)$. These function spaces are generalizations of normed vector spaces of functions which take values in $\mathbb{C}$. The thesis is split into two distinct areas of function space theory: analytic function spaces and harmonic analysis. There is, however, a common theme of matrix weights, particularly the reverse Hölder condition on matrix weights and a generalization of the $A_p$ conditions on matrix weights for $p=2$ and $p=\infty$.