Positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory, and they have recently been extended to the setting of free noncommutative function theory. In this paper, we develop the subject further in a number of directions. We give a characterization of completely positive noncommutative kernels in the setting of Hilbert C*-modules and Hilbert W*-modules. We prove an Arveson-type extension theorem for completely positive noncommutative kernels, and we show that a uniformly bounded noncommutative kernel can be decomposed into a linear combination of completely positive noncommutative kernels. / Ph. D. / Over the last several decades, positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory. Recently, they have been extended to the setting of free noncommutative function theory which is an active area of research with motivation from several different sources including free probability and noncommutative real semialgebraic geometry. In this paper, we develop further the theory of positive kernels in the noncommutative setting.