Vsevolod Alekseevich Solonnikov is known as one of the outstanding mathematicians from the St. Petersburg Mathematical School. His remarkable results on exact estimates of solutions to boundary and initial-boundary value problems for linear elliptic, parabolic, Stokes and Navier-Stokes systems, his methods and contributions to the inverstigation of free boundary problems, in particular in fluid mechanics, are well known to specialists all over the world. The International Conference on "Trends in Partial Differential Equations of Mathematical Physics" was held on the occasion of his 70th birthday in A"bidos (Portugal) from June 7 to 10, 2003. The conference consisted of thirty-eight invited and contributed lectures and gathered, in the charming and unique medieval town of A"bidos, about sixty participants from fifteen countries. This book contains twenty original contributions on many topics related to V.A. Solonnikov's work, selected from the invited talks of the conference.