<p>This paper applies a novel time-varying causality test as well as the wavelet coherence analysis, to investigate the nonlinear causal relationships between the cryptocurrency and the exchange rate markets, spanning the period January 3, 2017 to December 23, 2022. The time-varying causality test shows evidence of significant dynamic characteristics in the causal relationships between the cryptocurrency and the exchange rate markets for both the return and the volatility series across the sample period. Specifically, after the outbreak of the coronavirus disease 2019 (COVID-19) pandemic, a unidirectional causality emerges from the cryptocurrency market to the exchange rate market at the return level, while a bidirectional causality is observed between the two markets at the volatility level. Furthermore, the wavelet coherence analysis enables us to determine the direction of causality and the correlation in the time-frequency domain. The results reveal that in the short term, extreme events amplify the causal relationships between the two markets at both the return and volatility levels, though the direction of causality is capricious. In the middle and long term, a significantly positive causal relationship emerges from the cryptocurrency market to the exchange rate market at the return level, while a positive causal link flows in the opposite direction at the volatility level. The findings offer useful insights for investors, policymakers, and regulators and contribute to the literature on the cryptocurrency markets, thus, providing certain reference, for future studies in this area.</p>