Mathematics in the 20th century / Sir Michael Atiyah -- Function / N. Luzin -- Two letters from N.N. Luzin to M.Ya. Vygodskĭi -- Riemann's dissertation and its effect on the evolution of mathematics / Detlef Laugwitz -- Evolution of integration / A. Shenitzer and J. Steprāns -- On the calculus of variations and its major influences on the mathematics of the first half of our century / Erwin Kreyszig -- Evolution of literal algebra / I.G. Bashmakova and G.S. Smirnova -- Evolution of algebra 1800-1870 / I.G. Bashmakova and A.N. Rudakov -- What are algebraic integers and what are they for? / John Stillwell -- Genesis of the abstract ring concept / Israel Kleiner -- Field theory from equations to axiomatization / Israel Kleiner -- Elliptic curves / John Stillwell -- Modular miracles / John Stillwell -- Topology and abstract algebra as two roads of mathematical comprehension / Hermann Weyl -- Symmetry / Jacques Tits -- Exceptional objects / John Stillwell -- Problem of squarable lunes / M.M. Postnikov -- How hyperbolic geometry became respectable / Abe Shentizer -- Does mathematics distinguish certain dimensions of space? / Zdzisław Pogoda and Leszek M. Sokołowski --Glimpses of algebraic geometry / I.G. Bashmakova and E.I. Slavutin -- Four significant axiomatic systems and some of the issues associated with them / Stefan Mykytiuk and Abe Shenitzer -- Foundations of mathematics in the twentieth century / Victor W. Marek and Jan Mycielski -- Development of rigor in mathematical probability (1900-1950) / Joseph L. Doob -- Evolution of methods of convex optimization / V.M. Tikhomirov -- A few expository mini-essays / Abe Shenitzer -- On the emotional assumptions without which one could not effectively investigate the laws of nature / Vl.P. Vizgin -- Significance of mathematics: the mathematician's share in the general human condition / Wilhelm Magnus.