Cross-browser compatibility and responsive design are crucial aspects of frontend development. In my experience, I've ensured cross-browser compatibility by using CSS resets and vendor prefixes to maintain consistent styling across different browsers. I also utilize tools like BrowserStack to test and identify any discrepancies in how the application renders on various browsers. For responsive design, I follow a mobile-first approach, using CSS media queries to adapt the layout for different screen sizes. Additionally, I leverage frameworks like Bootstrap to streamline the process of creating responsive, grid-based layouts. This approach helps in delivering a seamless user experience across all devices and browsers.