Error control codes for real-time interactive applications such as audio and video streaming must operate under strict delay constraints and be resilient to burst losses. Previous works have characterized optimal streaming codes that guarantee perfect and timely recovery of all source packets when the burst loss is below a certain maximum threshold. In this work, we generalize the notion of streaming codes to the unequal error protection (UEP) setting. Toward this end, we define two natural notions of streaming codes; symbol-level UEP and packet-level UEP. In the symbol-level UEP, the symbols within each source packet have varying recoverability requirements. In the packet-level UEP scenario, packets at even time slots and odd time slots have different recovery guarantees. We discuss practical motivations for both settings and develop coding schemes. We establish optimality or near-optimality guarantees through information-theoretic converse bounds. Simulations over Gilbert and Fritchman channels show that our coding schemes outperform baseline schemes over a wide range of channel parameters.