In this thesis, we show that in affine /-space with / > 4, there are monomial curves with arbitrarily large minimal number of generators of the tangent cone and still having Cohen-Macaulay tangent cone. In order to prove this result, we give complete descriptions of the defining ideals of infinitely many families of monomial curves. We determine the tangent cones of these families of curves and check the Cohen-Macaulayness of their tangent cones by using Grobner theory. Also, we compute the Hilbert functions of these families of monomial curves. Finally, we make some genus computations by using the Hilbert polynomials for complete intersections in projective case and by using Riemann-Hurwitz formula for complete intersection curves of superelliptic type.