Lyusternik and Shnirel'man were the first to prove
Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has
at least three simple (non-self-intersecting), closed geodesics.
See, e.g., p.466 of Berger's *A Panoramic view of Riemannian Geometry*, or
this *Encyc.Math.* article.
(Apparently details in the L.-S. 1929 proof were not resolved until ~~1978~~ 1993.)

Q. Is there an extension to any Riemannian metric for $\mathbb{S}^3$? E.g., there exist at least $k > 1$ simple, closed geodesics in $\mathbb{S}^3$ under every Riemannian metric?

(*Added*.) As Igor Rivin points out, $k=1$ is known for any
smooth metric on $\mathbb{S}^n$ via a 1927 proof of Birkoff.

**. Much remains unclear to me, despite Igor's (extremely) useful citations. The bounty is offered for clarifying the situation (comments cited in several instances):**

*Added a bounty*(1) Klingenberg proves (*Mathematische Zeitschrift}*) in 1981 there are 4 closed geodesics on $\mathbb{S}^3$.

(2) Long&Duan prove (*Advances in Mathematics*) in 2009 that there at least 2 closed geodesics on $\mathbb{S}^3$—What happened in
the 28 yrs between? Was Klingenberg's proof not accepted, or did he prove something
nuancedly different?

(3) And it seems that none of these authors are addressing
*simplicity*, as @alvarezpaiva noted.
The L.&S. theorem explicitly proves non-self-intersection on $\mathbb{S}^2$.