Elliptic functions are double-periodic meromorphic functions on C, that means basically $$f(a+ib)=f((a+n)+i(b+m)) \qquad \forall a, b \in \mathbb{R},\; n, m \in \mathbb{Z}.$$ So they define a function on a torus. But what has this to do with an ellipse? To find the answer, we go on a journey to the origins of Riemann surfaces in the times of Euler and Lagrange and see the theory beautifully arising from the observation of integrals over some simple curves we already know from school.