Make up the website in one browser, looking how and what others. All at once would not consider, but in identifying specific problems to be fix.
Make sure vendorname prefixes were set. Recommend to install autoprefixer to the project, he will do it for you. If in doubt about some of the techniques/a css property, available on the website can i use
to see how these technologies are supported in different browsers
You can still check out the layout on the validity