Recent results in analytical fixed point theory

We survey recent results in analytical fixed point theory. Firstly, we state resolutions of long-standing problems in infinite dimensional topology-the Schauder conjecture, the compact AR problem, and the Banach problem on the Hilbert cube. Secondly, we list some fixed point theorems on Kakutani maps, generalized upper hemicontinuous maps, Fan-Browder maps, approximable maps, acyclic maps, and admissible or better admissible class B of maps. Some related results are also given.