Abstract: This thesis discusses the spectrum underlay paradigm in cognitive radio networks. In this paradigm cognitive radios, or secondary users, are allowed to transmit concurrently with the licensed users, or primary users, over the same channel. The main aim of this scheme is to enhance the utilization of the licensed bands. However, the secondary users may create excessive interference on the primary links, and this in turn would reduce the quality of service (QoS) of such links. For this reason, secondary users may transmit concurrently with the primary users under some interference constraint that guarantees a certain degree of QoS for the primary network. In this thesis, resource allocation for cognitive radios in spectrum underlay paradigm is considered in terms of admission and power control. Moreover, an information theoretic approach to characterize the achievable rates of such an underlay cognitive network is developed. In the resource allocation framework, an optimization problem that aims at maximizing the sum throughput of the secondary links under interference constraint on the primary receivers and minimum QoS requirements for each secondary link is considered. The problem is divided into two main subproblems. The first one deals with the feasibility of the network, in which an admission control algorithm that maximizes the number of admitted links into the network is proposed. Whereas, the second is the throughput maximization problem for the admitted links under the aforementioned constraints, where a solution based on sequential geometric iv programming (GP) is developed. The sum throughput maximization problem is then extended to consider power allocation of maximizing the overall some throughput of the primary and secondary networks. Afterwards, the potential achievable rates for a simple model of underlay cognitive networks (consists of a single secondary link and two primary links) have been characterized from an information theoretic perspective. The main approach is to establish an achievable rate region that depicts the sum rate of the primary network versus the throughput of the secondary link. The secondary link is assumed to perform ratesplitting so that the primary network can decode and cancel part of the interference created from the secondary link. Moreover, the secondary receiver is assumed to be able decode the signal of one primary transmitter and cancels its interference such that the sum primary rate is not affected. A necessary and sufficient condition on the satisfaction of the above constraint is introduced. Furthermore, the performance of rate-splitting by the secondary link is investigated in a Gaussian setup.