The book is aimed at developing and analyzing fundamentally new mathematical models of dynamicprocesses in the Earth's geospheres, taking into account heredity, in order to detect (predict) natural and manmadethreats. It is known that many natural, technical and social systems are characterized by the property ofheredity (heredity or memory), when the actual dynamic characteristics of the system are determined by itsentire prehistory. Mathematical formalization of such properties in mathematical models is usually ensuredby the introduction of integral terms (memory functionals) with difference kernels. Therefore, taking intoaccount the effects of heredity in dynamic processes and systems also determines the relevance of the study. The topic is devoted to the study of various dynamic processes in geospheres using mathematicalmodeling. Dynamic processes discussed in the book: geomagnetic dynamo, geoacoustic emission,microseismic vibrations, deformation processes, solar activity, seismic activity, processes of transferand accumulation of radon concentration. The book was the first to examine some dynamic processes that have hereditary effects. Researchmethods are based on the following sections of mathematics: mathematical and functional analysis,equations of mathematical physics, differential equations and their systems, integro-differentialequations, fractional calculus, vector analysis, computational mathematics, numerical methods. Alsomethods of object-oriented programming, parallel programming. It should be noted that the research used the principle of the model-algorithm-program triad.Therefore, one of the chapters of the book is devoted to computational algorithms with thepossibility of parallelization. The reader will become familiar with new models of geodynamic processes with heredity andapproaches to their study. The presented numerical algorithms can be reproduced or applied to othersimilar processes. The book will provide comprehensive information about the processes beingstudied and methods of their research, all necessary references will be provided, including thosedefining the current state of affairs in this issue.