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The discovery of the algorithm by Bruno Buchberger on July 1965, so-called the Buchberger algorithm, to compute Gröbner bases of ideals of the polynomial ring had led the birth of the exciting research area called 'Computer Algebra' in the modern mathematics. The 8th Mathematical Society of Japan Seasonal Institute (MSJ SI 2015) entitled 'The 50th Anniversary of Gröbner Bases' was held on July 2015, which is the 50th year following the discovery of the Buchberger algorithm. This volume is the proceedings of MSJ SI 2015 and consists of 14 papers related with computer algebra, algebraic statistics, D-modules, convex polytopes and toric ideals. These papers enable the reader to look over the current trends on Gröbner bases. Especially, young researchers can reach a treasury of fascinating research problems which are pending. Foreword was contributed by Bruno Buchberger, where a secret story on the discovery of the Buchberger algorithm is stated.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America