Floer homology was initiated by A. Floer around 40 years ago. Floer homology is infinite dimensional homology theory and is (in my opinion) the deepest theory among various infinite dimensional topology. Its applications are on symplectic geometry, gauge theory and mathematical physics.

I will survey various aspects on Floer homology from my point of view. The topics will include:

1) Basic concept and history:

The basic idea of Floer homology and how it has been developed especially in early days.

2) Foundation.

In general to give precise rigorous detail of Floer theory is hard. Many machinery and works have been developed for this purpose. I survey some of them.

3) Relation among various Floer theories.

4) Structures

In recent years many structures have been discovered and studied in Floer homology. They are very important in various applications and I will explain some of them.

5) Applications:

Application of Floer homology is important in symplectic geometry, gauge theory and low dimensional topology, and mathematical physics such as Mirror symmetry. I will explain some of them.

6) Challenge:

I want to mention some directions of the future research where Floer theory can develop.