We were fortunate. We studied under Arnold. We moved in his orbit and had the opportunity to discuss with him everything under the sun. For everyone of us this was a rare gift, a great good fortune in our lives.-From the Introduction. Leading mathematician and expert teacher, V.I.Arnold turns 60 in June of 1997. This volume contains a selection of original papers prepared for the occasion of this 60th anniversary by former students and other participants in Arnold's Moscow seminar. A weekly event since the mid-1960s, this seminar and its participants have been inspired by Arnold's creative ideas and universal approach to mathematics. The papers in this volume reflect Arnold's wide range of interests and his scientific contributions, including singularity theory, symplectic and contact geometry, mathematical physics, and dynamical systems. the spirit of this work is consistent with Arnold's view of mathematics, connecting different areas of mathematics and theoretical physics. The book is rich in applications and geometrical in nature.