Since the late 1970s, the speckle interference patterns ubiquitous in pulse-echo ultrasound images have been used to characterize sub-resolution tissue structure. During this time, new models, estimation methods, and processing techniques have proliferated, offering a wealth of recommendations for the task of tissue characterization. A literature review was performed to draw attention to these various methods and to critically track assumptions and gaps in knowledge. A total of 388 articles were collected from a systematic search for first-order speckle statistics in diagnostic ultrasound in the NIH PubMed database and Elsevier's Scopus database. Articles were grouped by basic characteristics and evaluated for addressing fundamental assumptions. A sampling of models and methods is presented in order to reveal the state of the art in speckle statistics as well as sources of measurement error and other important considerations. While this body of literature emphasizes the value of speckle analysis in diagnostic ultrasound, it is shown that relatively little attention is devoted to basic assumptions such as the linearity of system response and scatterer geometry. Additionally, several areas of investigation are available to improve upon speckle statistics analysis, potentially leading to the advancement of this unique tool.