The work by Maddah-Ali and Niesen demonstrated the benefits in reducing the transmission rate in a noiseless broadcast network by joint design of caching and delivery schemes. In their setup, each user learns the demands of all other users in the delivery phase. In this paper, we introduce the problem of demand private coded caching where we impose a privacy requirement that no user learns any information about the demands of other users. We provide an achievable scheme and compare its performance using the existing lower bounds on the achievable rates under no privacy setting. For this setting, when $N\leq K$ we show that our scheme is order optimal within a multiplicative factor of 8. Furthermore, when $N > K$ and $M\geq N/K$, our scheme is order optimal within a multiplicative factor of 4.