Abstract: This paper proposes a new class of unit quaternion curves in $SO(3)$. A general method is developed that transforms a curve in $R^3$ (defined as a weighted sum of basis functions) into its unit quaternion analogue in $SO(3)$. Applying the method to well-known spline curves (such as Bézier, Hermite, and B-spline curves), we are able to construct various unit quaternion curves which share many important differential properties with their original curves. Many of our naive common beliefs in geometry break down even in the simple non-Euclidean space $S^3$ or $SO(3)$. For example, the de Casteljau type construction of cubic B-spline quaternion curves does not preserve $C^2$-continuity [10]. Through the use of decomposition into simple primitive quaternion curves, our quaternion curves preserve most of the algebraic and differential properties of the original spline curves.