Some challenges in modern computer architecture include managing power consumption and heat dissipation, improving memory hierarchy and cache coherence, optimizing instruction execution and pipelining, dealing with multi-core and multi-threaded processors, improving performance and efficiency in parallel computing, addressing security concerns such as hardware vulnerabilities and attacks, managing data transfer and communication across different components and devices, and addressing the limitations of Moore's Law in scaling down transistor size and increasing processor speed. These challenges require constant research, innovation, and advancements in computer architecture to keep up with the evolving needs and demands of modern computing systems.