We study the sum degrees of freedom (DoF) of interference channels with hybrid beam-forming in which hybrid beam-forming composed of analog and digital precodings is employed at each node. For the two-user case, we completely characterize the sum DoF for an arbitrary number of antennas and RF chains by developing an achievable scheme optimized for the hybrid beam-forming structure and deriving its matching upper bound. For a general K-user case, we focus on a symmetric case and obtain lower and upper bounds on the sum DoF, which are tight for certain cases. The results show that hybrid beam-forming can increase the sum DoF of interference channel under certain conditions while it cannot improve the sum DoFs of point-to-point channel, multiple access channel, and broadcast channel. The key insights on this gain is that hybrid beam-forming enables users to manage inter user interference better, and thus each user can increase the dimension of interference-free signal space for its own desired signals.