Coping certain data sheet to sheet

I am needing help in copying certain information from 1 sheet to another. I have attached a sample spreadsheet and I want to copy certain information from ACE spreadsheet - only columns A-D not the whole row to the completed tab into the designated lists based on the criteria listed on the spreadsheet. Not sure if this is the most efficient way so any suggestions would be greatly appreciated.

Thanks in advance for your help and knowledge.

Hi again leem. Here is what I came up with:

Public Sub leem2()

Dim i, j, k, y, z As Integer
Dim FNextrow As Long
Dim MNextrow As Long
Dim LNextrow As Long
Dim SQtr As Date
Dim EQtr As Date
Dim lngDR As Long
Dim blnBlank As Boolean

Sheets("Completed").Activate
SQtr = CDate(InputBox("Please enter the first date of the current quarter"))
EQtr = CDate(InputBox("Please enter the final date of the current quarter"))
With Sheets("ACE")
    For i = 3 To .Cells(3, 1).End(xlDown).Row
        'Checks if ACE tab, column F=16.  If true, transfer to Completed sheet.
        FNextrow = Sheets("Completed").Cells(Rows.Count, 1).End(xlUp).Row + 1
        If .Cells(i, 6).Value = 16 Then
            Sheets("Completed").Range(Cells(FNextrow, 1), Cells(FNextrow, 5)).Value = .Range(.Cells(i, 1), .Cells(i, 5)).Value
        End If
        'Checks if ACE tab, column E=PS or E=TS, with other conditions listed below.
        MNextrow = Sheets("Completed").Cells(Rows.Count, 7).End(xlUp).Row + 1
        blnBlank = False
        Select Case .Cells(i, 5).Value
            Case "PS"
                For z = 7 To 37 Step 2
                    If .Cells(i, z).Value = 0 Then blnBlank = True
                Next z
                If blnBlank = False Then
                    Sheets("Completed").Range(Cells(MNextrow, 7), Cells(MNextrow, 11)).Value = .Range(.Cells(i, 1), .Cells(i, 5)).Value
                End If
            Case "TS"
                For y = 7 To 37 Step 2
                    If y <> 17 And y <> 19 And y <> 21 And y <> 33 And y <> 35 Then
                        If .Cells(i, y).Value = 0 Then blnBlank = True
                    End If
                Next y
                If blnBlank = False Then
                    Sheets("Completed").Range(Cells(MNextrow, 7), Cells(MNextrow, 11)).Value = .Range(.Cells(i, 1), .Cells(i, 5)).Value
                End If
        End Select
        lngDR = .Cells(Rows.Count, "AL").End(xlUp).Row
        LNextrow = Sheets("Completed").Cells(Rows.Count, "M").End(xlUp).Row + 1
        For j = 7 To 37 Step 2
            If .Cells(i, j).Value > SQtr Then
                If .Cells(i, j).Value < EQtr Then
                    Sheets("Completed").Range(Cells(LNextrow, 13), Cells(LNextrow, 17)).Value = .Range(.Cells(i, 1), .Cells(i, 5)).Value
                End If
            End If
        Next j
    Next i
End With

End Sub