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Manually testing all functionalities on an online store each time we deploy changes is really time-consuming, but on the other hand not doing so adds an additional opportunity for mistakes and bugs to happen like described above.
Each project should have at least smoke tests, which cover functionalities like:
Adding product to the cart
Checkout with different payment methods for logged in customers and guests
For functional testing, we are using Selenium with Jenkins. The combination of those tools allows us to run Smoke tests with one click on dev / staging / live environment.