Transient execution attacks that exploit speculation have raised significant concerns in computer systems. Typically, branch predictors are leveraged to trigger mis-speculation in transient execution attacks. In this work, we demonstrate a new class of speculation-based attack that targets branch prediction unit (BPU). We find that speculative resolution of conditional branches (i.e., in nested speculation) alter the states of pattern history table (PHT) in modern processors, which are not restored after the corresponding branches are later squashed. Such characteristic allows attackers to exploit BPU as the secret transmitting medium in transient execution attacks. To evaluate the discovered vulnerability, we build a novel attack framework, BranchSpectre, that enables exfiltration of unintended secrets through observing speculative PHT updates (in the form of covert and side channels). We further investigate PHT collision mechanism in the history-based predictor as well as the branch prediction mode transitions in Intel processors. Built upon such knowledge, we implement an ultra high-speed covert channel (BranchSpectre-cc) as well as two side channels (i.e., BranchSpectre-v1 and BranchSpectre-v2) that merely rely on BPU for mis-speculation trigger and secret inference in the speculative domain. Notably, BranchSpectre side channels can take advantage of much simpler code patterns than the ones used in Spectre attacks. We present an extensive BranchSpectre code gadget analysis on a set of popular real-world application code bases followed by a demonstration of real-world side channel attack on OpenSSL. The evaluation results show substantial wider existence and higher exploitability of BranchSpectre code patterns in real-world software. Finally, we discuss several secure branch prediction mechanisms that can mitigate transient execution attacks exploiting modern branch predictors.