Many computational problems on graphs are NP hard, so a natural strategy is to restrict them to some special graphs. This approach has seen many successes in the last few decades, and many efficient algorithms have been designed for problems on graph classes including graphs of bounded degree, bounded tree-width, and planar graphs, to name a few. As a matter of fact, many such algorithmic results can be understood in the framework of the so-called algorithmic meta-theorems. They are general results that provide efficient algorithms for decision problems of logic properties on structural graphs, which are also known as model-checking problems. Most existing algorithmic meta-theorems rely on modern structural graph theory, and they are often concerned with fixed-parameter tractable algorithms, i.e., efficient algorithms in the sense of parameterized complexity. On many well-structured graphs, the model-checking problems for some natural logics, e.g., first-order logic and monadic second-order logic, turn out to be fixed-parameter tractable. Due to varying expressive power, the tractability of the model-checking problems of those logics have huge differences as far as the underlying graph classes are concerned. Therefore, understanding the maximum graph classes that admit efficient model-checking algorithms is a central question for algorithmic meta-theorems. For example, it has been long known that efficient model-checking of first-order logic is closely related to the sparsity of input graphs. After decades of efforts, our understanding of sparse graphs are fairly complete now. So much of the current research has been focused on well-structured dense graphs, where challenging open problems are abundant. Already there are a few deep algorithmic meta-theorems proved for dense graph classes, while the research frontier is still expanding. This survey aims to give an overview of the whole area in order to provide impetus of the research of algorithmic meta-theorem in China.