Smart contracts are Turing-complete programs that are executed across a blockchain. Unlike traditional programs, once deployed, they cannot be modified. As smart contracts carry more value, they become more of an exciting target for attackers. Over the last years, they suffered from exploits costing millions of dollars due to simple programming mistakes. As a result, a variety of tools for detecting bugs have been proposed. Most of these tools rely on symbolic execution, which may yield false positives due to over-approximation. Recently, many fuzzers have been proposed to detect bugs in smart contracts. However, these tend to be more effective in finding shallow bugs and less effective in finding bugs that lie deep in the execution, therefore achieving low code coverage and many false negatives. An alternative that has proven to achieve good results in traditional programs is hybrid fuzzing, a combination of symbolic execution and fuzzing. In this work, we study hybrid fuzzing on smart contracts and present ConFuzzius, the first hybrid fuzzer for smart contracts. ConFuzzius uses evolutionary fuzzing to exercise shallow parts of a smart contract and constraint solving to generate inputs that satisfy complex conditions that prevent evolutionary fuzzing from exploring deeper parts. Moreover, ConFuzzius leverages dynamic data dependency analysis to efficiently generate sequences of transactions that are more likely to result in contract states in which bugs may be hidden. We evaluate the effectiveness of ConFuzzius by comparing it with state-of-the-art symbolic execution tools and fuzzers for smart contracts. Our evaluation on a curated dataset of 128 contracts and 21K real-world contracts shows that our hybrid approach detects more bugs (up to 23%) while outperforming state-of-the-art in terms of code coverage (up to 69%), and that data dependency analysis boosts bug detection up to 18%.