Recently, the studies on developing sensors and biosensors—with an obvious interdisciplinary character—have drawn the attention of many researchers specializing in various fundamental, but also complex domains such as chemistry, biochemistry, physics, biophysics, biology, bio-pharma-medicine, and bioengineering. Along these lines, the present paper is structured into three parts, and is aimed at synthesizing the most relevant studies on the construction and functioning of versatile devices, of electrochemical sensors and biosensors, respectively. The first part presents examples of the most representative scientific research focusing on the role and the importance of the phenylalanine, tyrosine, and tryptophan amino acids, selected depending on their chemical structure and their impact on the central nervous system. The second part is dedicated to presenting and exemplifying conductor polymers and molecularly imprinted polymers used as sensitive materials in achieving electrochemical sensors and biosensors. The last part of the review analyzes the sensors and biosensors developed so far to detect amino acids with the aid of conductor polymers and molecularly imprinted polymers from the point of view of the performances obtained, with emphasis on the detection methods, on the electrochemical reactions that take place upon detection, and on the electroanalytical performances. The present study was carried out with a view to highlighting, for the benefit of specialists in medicine and pharmacy, the possibility of achieving and purchasing efficient devices that might be used in the quality control of medicines, as well as in studying and monitoring diseases associated with these amino acids.