Cardiovascular diseases (including stroke and heart attack) are identified as the leading cause of death in today's world. However, very little is understood about the arterial mechanics of plaque buildup, arterial fibrous cap rupture, and the role of abnormalities of the vasa vasorum. Recently, ultrasonic echogenicity characteristics and morphological characterization of carotid plaque types have been shown to have clinical utility in classification of stroke risks. Furthermore, this characterization supports aggressive and intensive medical therapy as well as procedures, including endarterectomy and stenting. This is the first state-of-the-art review to provide a comprehensive understanding of the field of ultrasonic vascular morphology tissue characterization. This paper presents fundamental and advanced ultrasonic tissue characterization and feature extraction methods for analyzing plaque. Additionally, the paper shows how the risk stratification is achieved using machine learning paradigms. More advanced methods need to be developed which can segment the carotid artery walls into multiple regions such as the bulb region and areas both proximal and distal to the bulb. Furthermore, multimodality imaging is needed for validation of such advanced methods for stroke and cardiovascular risk stratification.