In this talk, asymptotic behavior of bounded solutions of almost periodic parabolic equation on S1 will be discussed.In particular, I will present some of our works on the structure of omega-limit sets. It was discovered that any minimal set can be residually embedded into an invariant set of almost automorphically-forced circle flow. In some special case, the flow on the minimal set topologically conjugates to an almost periodically forced flow on R or S^1. Furthermore, the structure of omega-limit sets will also be thoroughly investigated. Finally, I will prove that any non-wandering point of an autonomous or periodic scalar parabolic equation on S^1 should be a limit point.