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I recently made a flat surface and turned it into a rounded shape and sadly forgot on how that was achieved within Blender. the wished result is on the far right.

I tried to make it now with the "to sphere" function but that gives me really crappy results. I just know that I haven´t used the proportional editing nor a real sphere back then.

enter image description here

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up vote 3 down vote accepted

To sphere is actually the correct way if you do not want to use proportional editing, but you have to make sure that yout pivot center is set to where you want the center of the sphere to be.
The easiest way is probably to set the pivot point to be at the 3D Cursor
enter image description here
and then placing the 3D Cursor where you need it. You can either set its coordinates directly
enter image description here
or select a set of vertices or objects in whose center you want the cursor to be and press Shift+S and selecting 'Cursor to Selected'.
enter image description here

Btw: The radius of the sphere is the median distance of the selection to this pivot point.

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Starting with the left most object, delete the faces on the top, flat surface. Then switch to vertex select mode. Select all of the vertices around the top loop, extrude them, and join them at the center. Switch to edge select mode, select all of the edges from the center vertex to the edge, and subdivide he edges several (3 to five) times, depending upon the desired height of the dome. Form the dome by selecting all but the outermost edge loop, and raise the selected part to the desired height. Repeat the same process, selecting all but the outermost edge loop of the previous step, and finally the center vertex.

Alternatively, you can turn on proportional editing, if desired.

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