## 1 Introduction  
The development and operation of modern linear proton and heavy ion accelerators is impossible without the use of advanced computer programs to simulate the acceleration and evolution of a charged particle beam. For this purpose, a very original version of the multiparticle code DYNAMION [1] was created more than three decades ago at the Linac Division of the Institute for Theoretical and Experimental Physics (ITEP, Moscow, Russia). The algorithms used for the code are based on the theory of particle motion in linear accelerators including space charge effects, developed by the founder of the ITEP Linac Division, Prof. [PERSON] [2]. With the implementation of the DYNAMION software, various linac projects have been developed at ITEP [3, 4, 5, 6]. Over the years the code has been expanded within the framework of the long-standing international cooperation with GSI Helmholtzzentrum fuer Schwerionenforschung (GSI, Darmstadt, Germany) [7, 8].  
However, development of the next generation of high intensity linear accelerators requires a tool adequate to modern tasks. In particular, an advanced beam dynamics simulation software should be able to work with any known accelerating/focusing structures and their combination, taking all details of particle motion, caused by space charge effects, coupling between oscillations in different phase planes and nonlinearities of external fields into account. Also, the obtained calculated data should allow for deep and flexible analysis of the evolution of the particle ensemble. For this purpose, the software package DYNAION was thoroughly revised about twenty years ago. A high level of reliability was demonstrated by comparison of numerous measurement results obtained from operating linacs in several leading accelerator centers with corresponding simulations. Nevertheless, further code development is still in progress by adding different supplements in connection with the recent pioneer tasks.  
The presented software package stands among other well-known computer codes as Beampath [9], TRACK [10], TraceWin [11] and BEAMDULAC [12]. However, a number of purposely developed modeling features, especially in their entirety, allow to solve unique and complex tasks, making DYNAION code almost universal in the design and/or optimization of ion linear accelerators and decelerators.  
Over the last few decades, DYNAION has been widely used both for the development of new facilities and for the study and optimization of existing linacs at various accelerator centers around the world, namely at GSI (Darmstadt, Germany) [13; 14; 15; 16; 17; 18], ITEP (Moscow, Russia) [3; 4; 5; 6], HIM (Mainz, Germany) [19; 20], ANL (Argonne, IL, USA) [21], HIT (Heidelberg, Germany) [22] and INR (Troitsk, Russia) [23]. Supporting beam dynamics investigations have been performed for linac projects at CERN (Geneva, Switzerland) [24], LNL-INFN (Legnaro, Italy) [25], SOREQ (Tel-Aviv, Israel) [26] and JINR (Dubna, Russia) [27]. Some of these projects are briefly described in this manuscript, illustrating various capabilities of the code.  
A dedicated chapter is devoted to the design and successful commissioning of the RFQ-Decelerator for the HITRAP project at GSI [17].