This paper provides expressions and results for a family of saddle surfaces that includes the simple saddle and lesser-known monkey saddle as well as an infinite sequence of higher-order saddles that includes what could be called starfish and octopus saddles. Saddles often occur in real topography along drainage divides. Traversing a drainage divide involves repeatedly moving from a peak down a descending ridgeline to a saddle point and then up an ascending ridgeline to another peak. Peaks that are similar to monkey saddles also occur in real topography and are discussed. A derivation is given that provides polynomial expressions for an infinite family of high-order saddle surfaces. In addition, interesting general expressions are given for the plan, profile and streamline curvatures of these surfaces. While interesting on their own, these surfaces can also be used as test surfaces for geomorphometric analysis and algorithms.