This is a tough one, as for the longest time I didn’t believe in having a résumé as a designer. I still kind of believe that, as I think you should only really judge a designers work based on their portfolio.
However, as I got more experience in the industry, I discovered reputation is important, and a potential client knowing where you’ve worked before is too.
Also, it gives a quick overview of some details like where you are based and what level of agencies/studios you have worked with before. Not to forget, recruitment agencies need them. They will always ask for a résumé so it's good to have one sorted.
Keep it simple and at max two pages long. You don’t need to waffle too much, no date of birth, no personal hobbies and most importantly... DO NOT have a graph or chart showing the levels of your skills.
This is a big no as so many designers show their weaknesses in particular areas which don’t make you look good. Always stay on the positives, if you’re not amazing at Adobe Illustrator, don’t put it in.
Like your portfolio, keep it simple and clean, and updated! There are many free résumé mockup templates online, but here is a very basic breakdown of what I think needs to be showcased.