Meat quality parameters are affected by a complex series of interacting chemical, biochemical, physical, and physiological components that determine not only the suitability for consumption and the conditions for further processing and storage but also consumer acceptability. Deep understanding and careful manipulation of these intrinsic and extrinsic factors have to be taken in account to ensure high quality of meat, with better technological properties and increased safety for consumers. Among meat quality characteristics, meat tenderness has been perceived as the most important factor governing consumer acceptability. Therefore, being able to early predict meat texture and other related parameters in order to guarantee consistent eating quality to the final consumer is one of the most sought-after goals in the meat industry. Accurate measurements of both the biochemical and mechanical characteristics that underpin muscle and its transformation into meat are key factors to an improved understanding of meat quality, but also this early-stage measurements may be useful to develop methods to predict final meat texture. It is the goal of this review to present the available research literature on the historical and contemporary analyses that could be applied in early postmortem stages (pre-rigor and rigor) to determine the biochemical and physical characteristics of the meat that can potentially impact the eating quality.