In this thesis, we present an construction to all orientable closed surfaces and prove that any closed orientable surface may be smoothly embedded in Euclidean 3 - space so that when it is illuminated by parallel rays from any direction the shade cast on the surface is connected. In particular this problem gives counterexample of every genus to a conjecture of choe ([5]) states that any immersion of a surface of topological genus g should have at least one shade with g+1 components. Note that M has connected shades if and only if the guass map of M satisfies a two-pieceproperty (a set A in En is said to have the two-piece-property if and only if every hyperplane in En cuts A into at most to pieces). The study of shades is also of substantial interest in computer vision where ”shape fromshading” problems are studied extensively. One of basic questions, that considered by H.Wente in(1978), is, Does connectedness of each of the shades Su of a closed orientable surface M imply that M is convex? M.Ghomi showed that the answer is yes provided that either M is imply connected. otherwise, it was proved that the answer is no by constructing smooth embedded tori with connected shades. Also the first proposition in main chapter of this thesis, is concerned with the existence of closed curves without any pairs of parallel tangent lines, i.e., skew loops, and is an extension of a construction first discovered by B. Segre. It was shown that a tubular surface about a skew loop has connected shades. Here we show that one may construct a skew loop so that the corresponding tubular surface has any desired number of pairs of points which away from each other, we will then prove our second proposition which states that if a surface with connected shades has a pair of points p,q which face away from each other, then one may add a handle to that surface and thus increase its topological genus while preserving the connectedness of each of its shades. More precisely, we will delete small neighborhoods of p and q which are homeomorphic to disks and glue in their place a topological annulus. To this end we first deform neighborhoods of p and q until they coincide with pieces of spheres of the same radius, and then cut small disks from these spherical pieces. It will be shown that the resulting surface still has connected shades. We join the two boundary components of this surface with a surface of revolution which we call an hour glass. The hour glass has the crucial property that each component of each of its shades intersects its boundary. This implies that our final surface will have connected shades. Topology of surfaces with connected shades