Automated solutions for medical diagnosis based on computer vision form an emerging field of science aiming to enhance diagnosis and early disease detection. The detection and quantification of facial asymmetries enable facial palsy evaluation. In this work, a detailed review of the quantification of facial palsy takes place, covering all methods ranging from traditional manual mathematical modeling to automated computer vision-based methods. Moreover, facial palsy quantification is defined in terms of facial asymmetry indices calculation for different image modalities. The aim is to introduce readers to the concept of mathematical modeling approaches for facial palsy detection and evaluation and present the process of the development of this separate application field over time. Facial landmark extraction, facial datasets, and palsy grading systems are included in this research. As a general conclusion, machine learning methods for the evaluation of facial palsy lead to limited performance due to the use of handcrafted features, combined with the scarcity of the available datasets. Deep learning methods allow the automatic learning of discriminative deep facial features, leading to comparatively higher performance accuracies. Datasets limitations, proposed solutions, and future research directions in the field are also presented.