We introduce the SOC-convexity and SOC-monotonicity which are natural extensions of traditional convexity and monotonicity. These kinds of SOC-convex and SOC-monotone functions are also parallel to matrix-convex and matrix-monotone functions. We start with studying the SOC-convexity and SOC-monotonicity for some simple functions. Then, we explore characterizations of SOC-convex and SOC-monotone functions.