All sorts. For starters, no convex polygon with more than six sides tessellates, and various polyominoes of seven or more squares don't.
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No.
Yes it does tesselate.
It is possible, but not always true.
There are many shapes: circles, ellipses, ovals (elongated circles), cardioids, any shape with a "hole" in it such as a annulus.Polygons will tessellate if combined with other suitable polygons.
Yes, any quadrilateral will tessellate.