A cognitive transceiver is required to opportunistically use vacant spectrum resources licensed to primary users. Thus, it relies on a complete adaptive behavior composed of: reconfigurable radio frequency (RF) parts, enhanced spectrum sensing algorithms, and sophisticated machine learning techniques. In this paper, we present a review of the recent advances in CR transceivers hardware design and algorithms. For the RF part, three types of antennas are presented: UWB antennas, frequency-reconfigurable/tunable antennas, and UWB antennas with reconfigurable band notches. The main challenges faced by the design of the other RF blocks are also discussed. Sophisticated spectrum sensing algorithms that overcome main sensing challenges such as model uncertainty, hardware impairments, and wideband sensing are highlighted. The cognitive engine features are discussed. Moreover, we study unsupervised classification algorithms and a reinforcement learning (RL) algorithm that has been proposed to perform decision-making in CR networks.