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Database Management System is an umbrella term that refers to all sorts of completely different tools (i.e. computer programs or embedded libraries), mostly working in different and very unique ways. These applications handle, or heavily assist in handling, dealing with collections of information. Since information (or data) itself can come in various shapes and sizes, dozens of DBMS have been developed, along with tons of DB applications, since the second half of the 21st century to help in solving different programming and computerisation needs.
Database management systems are based on database models: structures defined for handling the data. Each emerging DBMS, and applications created to actualise their methods, work in very different ways with regards to definitions and storage-and-retrieval operations of said information.
Although there are a large number of solutions that implement different DBMs, each period in history has seen a relatively small amount of choices rapidly become extremely popular and stay in use for a longer time, with probably the most predominant choice since the past couple of decades (or even longer) being the Relational Database Management Systems (RDBMS).
Database Models
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Each database system implements a different database model to logically structure the data that is being managed. These models are the first step and the biggest determiner of how a database application will work and handle the information it deals with.
There are quite a few different types of database models which clearly and strictly provide the means of structuring the data, with most popular probably being the Relational Model.
Although the relational model and relational databases are extremely powerful and flexible - when the programmer knows how to use them, for many, there have been several issues or features that these solutions never