1. Metric Diophantine approximation – aspects of recent work Victor Beresnevich, Felipe Ramírez and Sanju Velani; 2. Exponents of Diophantine approximation Yann Bugeaud; 3. Effective equidistribution of nilflows and bounds on Weyl sums Giovanni Forni; 4. Multiple recurrence and finding patterns in dense sets Tim Austin; 5. Diophantine problems and homogeneous dynamics Manfred Einsiedler and Tom Ward; 6. Applications of thin orbits Alex Kontorovich; Index.