As we move into the NISQ era, a major challenge is quantum algorithms. In the short term this includes proposals for quantum supremacy and more broadly protocols for testing quantum systems. The key challenge of course is understanding the uses of these quantum computers, which involves exploring algorithms for quantum simulation, machine learning and optimization. Over the next few years before post-quantum cryptography is deployed, it will also be critically important to study quantum algorithms for breaking proposed post-quantum cryptosystems. And as experimentalists successfully understand the underlying noise models for qubits, the design of quantum error-correcting codes and protocols for fault-tolerant quantum computing will have to be revisited. I will talk about these challenges and key problems.