The Legacy of Vladimir Andreevich Steklov Nikolay Kuznetsov, Tadeusz Kulczycki, Mateusz Kwa´ snicki, Alexander Nazarov, Sergey Poborchi, Iosif Polterovich and Bartlomiej Siudeja Vladimir Andreevich Steklov, an outstanding Rus- sian mathematician whose 150th anniversary is celebrated this year, played an important role in the history of mathematics. Largely due to Steklov’s ef- forts, the Russian mathematical school that gave the world such giants as N. Lobachevsky, P. Chebyshev, and A. Lyapunov, survived the revolution and con- tinued to flourish despite political hardships. Steklov was the driving force behind the creation of the Physical–Mathematical Institute in starving Petro- grad in 1921, while the civil war was still raging in the newly Soviet Russia. This institute was the predecessor of the now famous mathematical insti- tutes in Moscow and St. Petersburg bearing Steklov’s name. Steklov’s own mathematical achievements, albeit less widely known, are no less remarkable than his contributions to the development of science. The Steklov eigenvalue problem, the Poincaré– Steklov operator, the Steklov function—there exist probably a dozen mathematical notions associated with Steklov. The present article highlights some of Nikolay Kuznetsov heads the Laboratory for Mathematical Modelling of Wave Phenomena at the Institute for Problems in Mechanical Engineering, Russian Academy of Sciences. His email address is nikolay.g.kuznetsov@gmail.com. Tadeusz Kulczycki and Mateusz Kwa´ snicki are professors at the Institute of Mathematics and Computer Science, Wroclaw University of Technology. Tadeusz Kulczycki’s email address is Tadeusz.Kulczycki@pwr.wroc.pl and Mateusz Kwa´ snicki’s email address is Mateusz.Kwasnicki@. pwr.wroc.pl. Alexander Nazarov is a leading researcher at the Labora- tory of Mathematical Physics, St. Petersburg Department of the Steklov Mathematical Institute, and a professor at the Department of Mathematical Physics, Faculty of Mathemat- ics and Mechanics, St. Petersburg State University. His email address is al.il.nazarov@gmail.com. His work was sup- ported by RFBR grant 11-01-00825 and by the St. Petersburg University grant 6.38.670.2013. DOI: http://dx.doi.org/10.1090/noti1073 V. A. Steklov in the 1920s. the milestones of his career, both as a researcher and as a leader of the Russian scientific community. Sergey Poborchi is a professor in the Department of Par- allel Algorithms, Faculty of Mathematics and Mechan- ics, St. Petersburg State University. His email address is poborchi@mail.ru. His work was supported by RFBR grant HK-11-01-00667/13. Iosif Polterovich is a professor in the Département de mathématiques et de statistique, Université de Montréal. His email address is iossif@dms.umontreal.ca. Bartlomiej Siudeja is a professor in the Department of Mathematics, University of Oregon. His email address is siudeja@uoregon.edu. The authors are grateful to David Sher for proofreading the article. January 2014 Notices of the AMS 9