This book collects original peer-reviewed contributions to the conferences organised by the international research network Minimal surfaces: Integrable Systems and Visualization financed by the Leverhulme Trust. The conferences took place in Cork, Granada, Munich and Leicester between 2016 and 2019. Within the theme of the network, the presented articles cover a broad range of topics and explore exciting links between problems related to the mean curvature of surfaces in homogeneous 3-manifolds, like minimal surfaces, CMC surfaces and mean curvature flows, integrable systems and visualisation. Combining research and overview articles by prominent international researchers, the book offers a valuable resource for both researchers and students who are interested in this research area.  

T. Bourni, M. Langford, and G. Tinaglia, Translating Solutions to Mean Curvature Flow.- A. Chern, F. Knoppel, F. Pedit, U. Pinkall and P. Schroder, Finding conformal and isometric immersions of surfaces.- J. Cho, W. Rossman, and S.-D Yang, Discrete minimal nets with symmetries.- B. Daniel, A survey on minimal isometric immersions into R3 , S2 X R and H2 X R.- L. Ferrer, F. Martn, R. Mazzeo, and M. Rodrguez.- Families of minimal surfaces in H2 X R foliated by arcs and their Jacobi fields.- E. S. Gama, E. Heinonen, J. H. de Lira and F. Martn, Jenkins-Serrin graphs in M X R.- E. Heinonen, Survey on the asymptotic Dirichlet problem for the minimal surface equation.- L. Heller, Generalized Whitham flow and its applications.- D. Hoffman, T. Ilmanen, F. Martn and B. White, Notes on translating solitons for Mean Curvature Flow.- M. Koiso, Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature and self-similar solutions of anisotropic mean curvature flow.- R. Lopez, The translating soliton equation.- E. Mota, Constant Mean Curvature Surfaces For The Bessel Equation.- A. C. Quintino, Minimal Surfaces under Constrained Willmore Transformation.- G. Smith, On an Enneper-Weierstrass-type representation of constant Gaussian curvature surfaces in 3-dimensional hyperbolic space.- P. M. Topping, Loss of initial data under limits of Ricci flows.- M. Yasumoto, Semi-discrete maximal surfaces with singularities in Minkowski space.