A surface M satisfying the Jacobi equation ?(f,g)= det =0 with respect to the Gaussian curvature K and the mean curvature H on a surface M is called a Weingarten surface or a W-surface. Also, if surface M satisfies a linear equation with respect to K and H, that is, aK + bH = c , ( a,b,c )? 0, a,b,c R then it is said to be a linear Weingarten surface or a LW-surface. If the second fundamental form II of a surface M in E^3 is nondegenerate, then it is regarded as a new Riemannian metric. Therefore, new Gaussian curvature and new mean curvature for Riemannian manifold ( M,II ) can be defined that call respectively second Gaussian curvature on M or K_ II and second mean curvature on M or H _ II . In this thesis, at chapter 3, first we study the Gaussian curvature, the mean curvature, the second Gaussian curvature on a surface M . Also, for a minimal surface M, we show that a minimal surface has a vanishing second Gaussian curvature, but a surface with the vanishing second Gaussian curvature need not to be a minimal surface. In the second step, we give the relationship between nondegenerate second fundamental form and non-zero Gaussian curvature in this case i.e., the second fundamental form of surface M is nondegenerate if and only if the Gaussian curvature of surface M never vanishes. By use of this fact, we give the second mean curvature of surface M . Let ? : [ a,b ] E^3 be a unit-speed regular curve. In the chapter 4, we study the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature on a tubular surface M, Furthermore, if ( X,Y ) {( K,H_ II ) , ( H,H_ II ) , ( H_ II, K_ II ) }, and suppose that M is a tubular surface with nondegenerate second fundamental form. Then, M is a ( X,Y ) - Weingarten surface if and only if M is a tubular surface around a circle or a helix. In chapter 5, we study linear Weingarten tubular surfaces in E^3 and it is shown that there are no ( K,H ) , ( K,K_ II ) , ( H,K_ II ) , ( K,H_ II ) , ( H,H_ II ) , ( K_ II, H_ II ) , ( H,K_ II, H_ II ) , ( K,K_ II, H_ II ) , ( K,H,H_ II ) , ( K,H,K_ II ) and ( K,K_ II, H_ II ) linear Weingarten tubular surfaces M in Euclidean3-space with nondegenerate second fundamental form. . In last chapter of this thesis, we show that if M is a Weingarten quadric surface in E^3, then M is an open part of one of a hyperboloid of two sheets, a hyperboloid of one sheet, an ellipsoid or an elliptic paraboloid.