This book contains a collection of lectures presented at the 2013 CIMPA-UNESCOSouth Africa School Evolutionary Equations with Applications in Natural Sciences. The School was part of the initiative Mathematics of Planet Earth and it was organized at the African Institute of Mathematical Sciences, http://www.aims.ac. za/, in Muizenberg (Cape Town), South Africa, from 22nd July to 2nd August 2013, under the auspices of the Centre International de Mathématiques Pures et Appliquées (CIMPA). CIMPA is a non-profit international organization whose aim is to promote international cooperation in higher education and research in mathematics for the benefit of developing countries. In accordance with CIMPA’s mission, the School was aimed at postgraduate students and young researchers from such countries. The School consisted of 9 courses delivered by invited lecturers coming from France, Germany, Poland, South Africa and Scotland, and attracted 29 participants from countries as diverse as Algeria, Benin, Cameroon, Chad, Congo, DRC, Ethiopia, Ivory Coast, Kenya, Lesotho, Mauritania, Morocco, Nigeria, Philippines, Pakistan, Sudan, South Africa, and Zimbabwe.