This synopsis represents an overview of some recent developments, explorations and extensions of input-output analysis undertaken during last decade in the Regional Economics Application Laboratory, University of Illinois at Urbana-Champaign, USA and the Bar-Ilan Laboratory of Environmental Information and Dynamics, Bar-Ilan University, Israel. This paper is the first of three review papers; in Part I, the focus will be on structure and structural changes in input-output systems and the following issues will be considered: I. Temporal multipliers, temporal increment and temporal Leontief inverse: fields of influence of changes. II. Economic Hierarchical Landscapes of backward and forward linkages: Key Sector Analysis, fields of influence and Multiplier Product Matrix. III. Economic structural changes in the input-output systems: Barycentric Calculus of triple decompositions of gross output changes. IV. Structural Q-analysis of the Leontief inverse. In Parts II and III, attention will be directed to multiregional input-output analysis and the transition from comparative statics to dynamics.