An Introduction to Mathematical Population Dynamics : Along the trail of Volterra and Lotka This book is an introduction to mathematical biology for students with no experience in biology but who have some mathematical background. The work is focused on population dynamics and ecology following a tradition that goes back to Lotka and Volterra and includes a part devoted to the spread of infectious diseases a field where mathematical modeling is extremely popular. These themes are used as the area where to understand different types of mathematical modeling and the possible meaning of qualitative agreement of modeling with data. The book also includes a collections of problems designed to approach more advanced questions. This material has been used in the courses at the University of Trento directed at students in their fourth year of studies in Mathematics. It can also be used as a reference as it provides up to date developments in several areas 1 Malthus Verhulst and all that 2 Delayed population models 3 Models of discrete time population growth 4 Stochastic modeling of population growth 5 Spatial spread of a population 6 Prey predator models 7 Competition among species 8 Mathematical modeling of epidemics 9 Models with several species and trophic levels 10 Appendices: A Basic theory of Ordinary Differential Equations; B Delay Equations; C Discrete dynamics; D Continuous time Markov chains