Simulation results show that the performance of polar codes is improved vastly by using polar codes as inner codes in serially concatenated coding schemes. Furthermore, this performance improvement is achieved using a relatively short cyclic redundancy check as the outer code and a practically implementable successive cancellation list decoder for decoding the overall code. This paper offers a theoretical analysis of such schemes by employing a random-coding method on the selection of the outer code and assuming that the channel is memoryless. It is shown that the probability of error for the concatenated coding scheme decreases exponentially in the code block length at any fixed rate below the symmetric capacity. Applications of this result include the design of polar codes for communication systems that require high reliability at small to moderate code lengths, such as control channels in wireless systems and machine-type communications for industrial automation.