María Barbero Liñán is an associate professor in the department of applied mathematics at Universidad Politécnica de Madrid, Spain. After obtaining her Ph.D. in Applied Mathematics at Universidad Politécnica de Cataluña, she has been postdoctoral researcher at INRIA (Nancy, France), Queen's University (Kingston, ON, Canada), ICMAT and Universidad Carlos III de Madrid in Spain.Kurusch Ebrahimi-Fard is an associate professor of mathematics in the department of mathematical sciences at the Norwegian University of Science and Technology in Trondheim, Norway. After obtaining his Ph.D. in Theoretical Physics from Bonn University, he has held postdoctoral positions, among others, at the IHES (Bures-Sur-Yvette, France), Max Planck Institute for Mathematics (Bonn, Germany) and the Instituto de Ciencias Matemáticas (Madrid, Spain).

• Preface.- A. Iserles and G.R.W. Quispel, Why geometric numerical integration?.- B. Owren, Lie group integrators.- H. Z. Munthe-Kaas and K. K. Føllesdal, Lie–Butcher series, Geometry, Algebra and Computation.- A. Murua and J. M. Sanz-Serna, Averaging and computing normal forms with word series algorithms.- L. A. Duffaut Espinosa, K. Ebrahimi-Fard, and W. Steven Gray, Combinatorial Hopf algebras for interconnected nonlinear input-output systems with a view towards discretization.- F. Casas, Computational aspects of some exponential identities.- K. Ebrahimi-Fard and I. Mencattini, Post-Lie Algebras, Factorization Theorems and Isospectral Flows.- G. Bogfjellmo, R. Dahmen, and A.Schmeding, Overview of (pro-)Lie group structures on Hopf algebra character groups,.- M. Barbero Liñán and D. Martín de Diego, Bäcklund transformations in discrete variational principles for Lie–Poisson equations.- M. Vermeeren, Numerical precession in variational discretizations of the Kepler problem.- O. Verdier,  Full affine equivariance and weak natural transformations in numerical analysis - the case of B-Series.- References.

María Barbero Liñán is an associate professor in the department of applied mathematics at Universidad Politécnica de Madrid, Spain. After obtaining her Ph.D. in Applied Mathematics at Universidad Politécnica de Cataluña, she has been postdoctoral researcher at INRIA (Nancy, France), Queen's University (Kingston, ON, Canada), ICMAT and Universidad Carlos III de Madrid in Spain.Kurusch Ebrahimi-Fard is an associate professor of mathematics in the department of mathematical sciences at the Norwegian University of Science and Technology in Trondheim, Norway. After obtaining his Ph.D. in Theoretical Physics from Bonn University, he has held postdoctoral positions, among others, at the IHES (Bures-Sur-Yvette, France), Max Planck Institute for Mathematics (Bonn, Germany) and the Instituto de Ciencias Matemáticas (Madrid, Spain).

• Preface.- A. Iserles and G.R.W. Quispel, Why geometric numerical integration?.- B. Owren, Lie group integrators.- H. Z. Munthe-Kaas and K. K. Føllesdal, Lie–Butcher series, Geometry, Algebra and Computation.- A. Murua and J. M. Sanz-Serna, Averaging and computing normal forms with word series algorithms.- L. A. Duffaut Espinosa, K. Ebrahimi-Fard, and W. Steven Gray, Combinatorial Hopf algebras for interconnected nonlinear input-output systems with a view towards discretization.- F. Casas, Computational aspects of some exponential identities.- K. Ebrahimi-Fard and I. Mencattini, Post-Lie Algebras, Factorization Theorems and Isospectral Flows.- G. Bogfjellmo, R. Dahmen, and A.Schmeding, Overview of (pro-)Lie group structures on Hopf algebra character groups,.- M. Barbero Liñán and D. Martín de Diego, Bäcklund transformations in discrete variational principles for Lie–Poisson equations.- M. Vermeeren, Numerical precession in variational discretizations of the Kepler problem.- O. Verdier,  Full affine equivariance and weak natural transformations in numerical analysis - the case of B-Series.- References.

María Barbero Liñán is an associate professor in the department of applied mathematics at Universidad Politécnica de Madrid, Spain. After obtaining her Ph.D. in Applied Mathematics at Universidad Politécnica de Cataluña, she has been postdoctoral researcher at INRIA (Nancy, France), Queen's University (Kingston, ON, Canada), ICMAT and Universidad Carlos III de Madrid in Spain.Kurusch Ebrahimi-Fard is an associate professor of mathematics in the department of mathematical sciences at the Norwegian University of Science and Technology in Trondheim, Norway. After obtaining his Ph.D. in Theoretical Physics from Bonn University, he has held postdoctoral positions, among others, at the IHES (Bures-Sur-Yvette, France), Max Planck Institute for Mathematics (Bonn, Germany) and the Instituto de Ciencias Matemáticas (Madrid, Spain).

• Preface.- A. Iserles and G.R.W. Quispel, Why geometric numerical integration?.- B. Owren, Lie group integrators.- H. Z. Munthe-Kaas and K. K. Føllesdal, Lie–Butcher series, Geometry, Algebra and Computation.- A. Murua and J. M. Sanz-Serna, Averaging and computing normal forms with word series algorithms.- L. A. Duffaut Espinosa, K. Ebrahimi-Fard, and W. Steven Gray, Combinatorial Hopf algebras for interconnected nonlinear input-output systems with a view towards discretization.- F. Casas, Computational aspects of some exponential identities.- K. Ebrahimi-Fard and I. Mencattini, Post-Lie Algebras, Factorization Theorems and Isospectral Flows.- G. Bogfjellmo, R. Dahmen, and A.Schmeding, Overview of (pro-)Lie group structures on Hopf algebra character groups,.- M. Barbero Liñán and D. Martín de Diego, Bäcklund transformations in discrete variational principles for Lie–Poisson equations.- M. Vermeeren, Numerical precession in variational discretizations of the Kepler problem.- O. Verdier,  Full affine equivariance and weak natural transformations in numerical analysis - the case of B-Series.- References.