MADGRAPH 5 is a new version of the matrix element generator written in Python. It introduces improved algorithms for enhanced performance and functionality. It features a new user interface, new output formats including C++ libraries for PYTHIA 8, and full compatibility with FEYNRules for new physics models. MADGRAPH 5 supports collaborative development and allows theoretical, phenomenological, and simulation projects to be developed and distributed. It provides a wide range of output formats in Fortran, C++, and Python, and dedicated matrix element output for PYTHIA 8. The paper describes the general philosophy and key improvements of MADGRAPH 5. It covers diagram generation, helicity amplitude call generation, colour algebra, and decay chains. It also discusses the validation and speed benchmarks, BSM example applications, and the new features of the code. The paper is structured into sections covering the overview and algorithms, outputs, models, validation, and examples. It also includes appendices with technical details and examples. MADGRAPH 5 is open source and written in Python, allowing for efficient and flexible algorithm implementation. It supports Lagrangian-based BSM physics via FEYNRules, full automation of NLO computations, and merging to showering/hadronization codes for complete event simulation. It can generate matrix elements for any Lagrangian-based model implemented in FEYNRules via the UFO interface and automatically generate helicity amplitude subroutines via the ALOHA package. It features a wide set of flexible output formats and efficient generation of decay chains. The diagram generation algorithm in MADGRAPH 5 is faster and more efficient than previous versions. It uses model information to construct valid diagrams and recursively generates diagrams in parallel. It also introduces a flag to avoid double-counting diagrams. The helicity amplitude call generation and fermion number violation are handled with special care for Majorana particles and fermion number violating vertices. The colour algebra module is implemented symbolically, allowing for flexibility and efficiency in handling complex colour structures. The decay chain generation is defined using successive chains of processes, allowing for quick and efficient generation of decay chains of virtually unlimited length. The outputs include matrix elements in Fortran 77, C++, and Python, and diagrams of Feynman diagrams. The paper also discusses the validation and speed benchmarks, BSM example applications, and the new features of the code.MADGRAPH 5 is a new version of the matrix element generator written in Python. It introduces improved algorithms for enhanced performance and functionality. It features a new user interface, new output formats including C++ libraries for PYTHIA 8, and full compatibility with FEYNRules for new physics models. MADGRAPH 5 supports collaborative development and allows theoretical, phenomenological, and simulation projects to be developed and distributed. It provides a wide range of output formats in Fortran, C++, and Python, and dedicated matrix element output for PYTHIA 8. The paper describes the general philosophy and key improvements of MADGRAPH 5. It covers diagram generation, helicity amplitude call generation, colour algebra, and decay chains. It also discusses the validation and speed benchmarks, BSM example applications, and the new features of the code. The paper is structured into sections covering the overview and algorithms, outputs, models, validation, and examples. It also includes appendices with technical details and examples. MADGRAPH 5 is open source and written in Python, allowing for efficient and flexible algorithm implementation. It supports Lagrangian-based BSM physics via FEYNRules, full automation of NLO computations, and merging to showering/hadronization codes for complete event simulation. It can generate matrix elements for any Lagrangian-based model implemented in FEYNRules via the UFO interface and automatically generate helicity amplitude subroutines via the ALOHA package. It features a wide set of flexible output formats and efficient generation of decay chains. The diagram generation algorithm in MADGRAPH 5 is faster and more efficient than previous versions. It uses model information to construct valid diagrams and recursively generates diagrams in parallel. It also introduces a flag to avoid double-counting diagrams. The helicity amplitude call generation and fermion number violation are handled with special care for Majorana particles and fermion number violating vertices. The colour algebra module is implemented symbolically, allowing for flexibility and efficiency in handling complex colour structures. The decay chain generation is defined using successive chains of processes, allowing for quick and efficient generation of decay chains of virtually unlimited length. The outputs include matrix elements in Fortran 77, C++, and Python, and diagrams of Feynman diagrams. The paper also discusses the validation and speed benchmarks, BSM example applications, and the new features of the code.