Monotone operators play important roles in optimization and convex analysis. We define a new average of monotone operators by using resolvents. The new average enjoys self-duality and inherits many nice features of given monotone operators. When the monotone operators are positive definite matrices, the new average lies between the harmonic average and arithmetic average. Appropriate limits of resolvent average lead to both harmonic average and arithmetic average. Consequences on matrix functions are also given.