Are submanifolds of $\mathbb{R}^n$ cut out by smooth functions?

Question

Let $M$ be a submanifold of $\mathbb{R}^n$. I was wondering can one always find (perhaps finite) smooth functions that cut out $M$? I was wondering about analogies between manifolds and (affine) varieties and was not sure if this holds or not. Any comments would be appreciated. Thank you.