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Private information retrieval (PIR) allows users to retrieve data from databases without revealing the identity of that data. An extensive body of works has investigated efficient schemes to achieve computational and information-theoretic privacy. The latter guarantees that no information is revealed to the databases, irrespective of their computational power. Although information-theoretic PIR (IT-PIR) provides a strong privacy guarantee, it can be too taxing for certain applications. In this paper, we initiate the study of leaky private information retrieval (L-PIR), where a bounded amount of privacy leakage is allowed and measured through a parameter ε. The classical IT-PIR formulation is obtained by setting ε = 0, and for ε > 0, we explore the opportunities offered for reducing the download cost. We derive new upper and lower bounds on the download cost of L-PIR for any arbitrary ε, any number of messages K, and for N = 2 databases.