Temporal processing is fundamental for both biological and artificial intelligence systems, as it enables the comprehension of dynamic environments and facilitates timely responses. Spiking Neural Networks (SNNs) excel in handling such data with high efficiency, owing to their rich neuronal dynamics and sparse activity patterns. Given the recent surge in the development of SNNs, there is an urgent need for a comprehensive evaluation of their temporal processing capabilities. In this paper, we first conduct an in-depth assessment of commonly used neuromorphic benchmarks, revealing critical limitations in their ability to evaluate the temporal processing capabilities of SNNs. To bridge this gap, we further introduce a benchmark suite consisting of three temporal processing tasks characterized by rich temporal dynamics across multiple timescales. Utilizing this benchmark suite, we perform a thorough evaluation of recently introduced SNN approaches to elucidate the current status of SNNs in temporal processing. Our findings indicate significant advancements in recently developed spiking neuron models and neural architectures regarding their temporal processing capabilities, while also highlighting a performance gap in handling long-range dependencies when compared to state-of-the-art non-spiking models. Finally, we discuss the key challenges and outline potential avenues for future research.