Due to their fundamental importance for premixed combustion theory, turbulent flame speed and thickness were a subject of a large number of investigations for many decades. The paper reviews the research and extensively discusses the still unresolved issues in an attempt to define a foundation for evaluating different combustion models and defining a simple approach to multi-dimensional computations of premixed turbulent combustion. The approach consists of the use of an algebraic expression for the local turbulent flame speed in order to close the averaged balance equations describing the combustion process. Several models have been suggested utilizing this approach and the first laboratory and industrial applications of them have shown encouraging results. These successful applications motivate a thorough discussion and further development of the approach. Models utilizing the approach are reviewed and two issues are emphasized. First, certain models focus on the combustion regime characterized by a growing mean flame brush thickness, whereas other models are associated with fully developed flames of asymptotically stationary structure. Second, many different expressions for flame speed are invoked by different models. Thus, the behavior of the mean flame brush thickness and flame speed should be analyzed in order to provide a more solid phenomenological base for the approach and this is the main goal of the paper. Moreover, such an analysis also aims at selecting experimentally well-established trends and, thus, contributes to the development of a database necessary for testing various models of premixed turbulent combustion. Sources of errors in measurements of turbulent flame speeds are discussed and strong quantitative scatter of the published data is demonstrated. Nevertheless, turbulent flame speed, S, is shown to be a phenomenologically meaningful quantity, because various experimental investigations indicate the same qualitative trends in the behavior of S-t at moderate turbulence, The following trends: (1) an increase in S-t by rms turbulent velocity u'; (2) an increase in S-t and dS(t)/du' by the laminar burning velocity; and (3) an increase in S-t by pressure despite the decrease in the laminar burning velocity, are well-established and can be used for testing various models of premixed turbulent combustion. Moreover, certain experimental results indicate a decrease in S-t by molecular transfer coefficients, other things being equal, and this trend may also be used for testing models. A number of various expressions for S-t, available in the literature, are tested against well-established trends, but only a few expressions are shown to be able to predict all the basic trends. An analysis of numerous experimental data obtained by various teams under different conditions indicates that a self-similar regime of premixed turbulent combustion, characterized by growing mean flame brush thickness, delta(t), arid by the universal dimensionless spatial profile of the progress variable across the brush, occurs in most laboratory and industrial burners. The development of delta(t) is mainly controlled by turbulent diffusion. Only certain models are able to describe this regime. From the group of models evaluated positively, the Flame Speed Closure (FSC) model is highlighted since: (1) it corresponds to the regime of growing mean flame brush thickness; and (2) it utilizes an experimentally well-supported expression for turbulent flame speed. Various numerical tests of the model, performed by numerous teams under substantially different conditions are summarized. Further development and validation of the model and it applications are reviewed. Finally, the paper shows that, after decades of long research, a simple, robust, conceptually straightforward, and extensively validated premixed combustion simulation tool is available for applications now.