Metamodeling
A metamodel is a model of a model, and metamodeling is the process of generating such metamodels. Thus metamodeling or meta-modeling is the analysis, construction and development of the frames, rules, constraints, models and theories applicable and useful for modeling a predefined class of problems. As its name implies, this concept applies the notions of meta- and modeling in software engineering and systems engineering. Metamodels are of many types and have diverse applications. Overview 
A metamodel/ surrogate model is a model of the model, i.e. a simplified model of an actual model of a circuit, system, or software like entity. Metamodel can be a mathematical relation or algorithm representing input and output relations. A model is an abstraction of phenomena in the real world;  a metamodel is yet another abstraction, highlighting properties of the model itself. A model conforms to its metamodel in the way that a computer program conforms to the grammar of the programming language in which it is written. Various types of metamodels include polynomial equations, neural network, Kriging, etc. "Metamodeling" is the construction of a collection of "concepts" (things, terms, etc.) within a certain domain. Metamodeling typically involves studying the output and input relationships and then fitting right metamodels to represent that behavior. Common uses for metamodels are:
 As a schema for semantic data that needs to be exchanged or stored
 As a language that supports a particular method or process
 As a language to express additional semantics of existing information
 As a mechanism to create tools that work with a broad class of models at run time
 As a schema for modeling and automatically exploring sentences of a language with applications to automated test synthesis
 As an approximation of a higher-fidelity model for use when reducing time, cost, or computational effort is necessary

Because of the "meta" character of metamodeling, both the praxis and theory of metamodels are of relevance to metascience, metaphilosophy, metatheories and systemics, and meta-consciousness. The concept can be useful in mathematics, and has practical applications in computer science and computer engineering/software engineering. The latter are the main focus of this article. Topics

Definition 
In software engineering, the use of models is an alternative to more common code-based development techniques. A model always conforms to a unique metamodel. One of the currently most active branch of Model Driven Engineering is the approach named model-driven architecture proposed by OMG. This approach is embodied in the Meta Object Facility (MOF) specification. Typical metamodelling specifications proposed by OMG are UML, SysML, SPEM or CWM. ISO has also published the standard metamodel ISO/IEC 24744. All the languages presented below could be defined as MOF metamodels. Metadata modeling 
Metadata modeling is a type of metamodeling used in software engineering and systems engineering for the analysis and construction of models applicable and useful to some predefined class of problems. (see also: data modeling). Model transformations 
One important move in model-driven engineering is the systematic use of model transformation languages. The OMG has proposed a standard for this called QVT for Queries/Views/Transformations. QVT is based on the meta-object facility (MOF). Among many other model transformation languages (MTLs), some examples of implementations of this standard are AndroMDA, VIATRA, Tefkat, MT, ManyDesigns Portofino. Relationship to ontologies 
Meta-models are closely related to ontologies. Both are often used to describe and analyze the relations between concepts
 Ontologies: express something meaningful within a specified universe or domain of discourse by utilizing a grammar for using vocabulary. The grammar specifies what it means to be a well-formed statement, assertion, query, etc. (formal constraints) on how terms in the ontology’s controlled vocabulary can be used together. Meta-modeling: can be considered as an explicit description (constructs and rules) of how a domain-specific model is built. In particular, this comprises a formalized specification of the domain-specific notations. Typically, metamodels are – and always should follow - a strict rule set. "A valid metamodel is an ontology, but not all ontologies are modeled explicitly as metamodels". Types of metamodels 
For software engineering, several types of models (and their corresponding modeling activities) can be distinguished:
Metadata modeling (MetaData model)
Meta-process modeling (MetaProcess model)
Executable meta-modeling (combining both of the above and much more, as in the general purpose tool Kermeta)
Model transformation language (see below)
Polynomial metamodels
Neural network metamodels
Kriging metamodels
Piecewise polynomial (spline) metamodels
Gradient-enhanced kriging (GEK)

Zoos of metamodels 
A library of similar metamodels has been called a Zoo of metamodels. There are several types of meta-model zoos. Some are expressed in ECore. Others are written in MOF 1.4 – XMI 1.2. The metamodels expressed in UML-XMI1.2 may be uploaded in Poseidon for UML, a UML CASE tool. See also 

 Business reference model
 Data governance
 Model-driven engineering (MDE)
 Model-driven architecture (MDA)
 Domain Specific Language (DSL)
 Domain-Specific Modeling (DSM)
 Generic Eclipse Modeling System (GEMS)
 Kermeta (Kernel Meta-modeling)
 Metadata
 MetaCASE tool (tools for creating tools for computer-aided software engineering tools)
 Method engineering
 MODAF Meta-Model
 MOF Queries/Views/Transformations (MOF QVT)
 Object Process Methodology
 Requirements analysis
 Space mapping
 Surrogate model
 Transformation language
 VIATRA (Viatra)
 XML transformation language (XML TL)

References

Further reading 

 
 Booch, G., Rumbaugh, J., Jacobson, I. (1999), The Unified Modeling Language User Guide, Redwood City, CA: Addison Wesley Longman Publishing Co., Inc.
 J. P. van Gigch, System Design Modeling and Metamodeling, Plenum Press, New York, 1991
 Gopi Bulusu, hamara.in, 2004 Model Driven Transformation
 P. C. Smolik, Mambo Metamodeling Environment, Doctoral Thesis, Brno University of Technology. 2006
 Gonzalez-Perez, C. and B. Henderson-Sellers, 2008. Metamodelling for Software Engineering. Chichester (UK): Wiley. 210 p. 
 M.A. Jeusfeld, M. Jarke, and J. Mylopoulos, 2009. Metamodeling for Method Engineering. Cambridge (USA): The MIT Press. 424 p. , Open access via http://conceptbase.sourceforge.net/2021_Metamodeling_for_Method_Engineering.pdf
 G. Caplat Modèles & Métamodèles,  2008  -   
 Fill, H.-G., Karagiannis, D., 2013. On the Conceptualisation of Modelling Methods Using the ADOxx Meta Modelling Platform, Enterprise Modelling and Information Systems Architectures, Vol. 8, Issue 1, 4-25. Software design
Scientific modelling