We know that convex lens converge light at a point,  
Is there any other shapes that could also converge light?

What if a convex len has only one side circular while the other end is flattened, when the incident light enters the circular side, what will be the transmitted light?

Same as glasses for long-sight. They are called plano-convex lenses: 

https://www.firebirdoptics.com/blog/intro-to-the-plano-convex-lens#:~:text=A plano-convex lens is,the lens being bowed outward.

A circle (or sphere) is not, in general, the right shape to  focus parallel light to a point.
https://en.wikipedia.org/wiki/Spherical_aberration

 

No. See Fresnel lens.

A flat lens will bend light, if the index of refraction varies with position. GRadient INdex, or GRIN lenses

 

3 hours ago, Genady said:

No. See Fresnel lens.

That still relies on curved surfaces

6 hours ago, kenny1999 said:

What if a convex len has only one side circular while the other end is flattened, when the incident light enters the circular side, what will be the transmitted light?

These are called plano-convex (as opposed to bi-convex, curved on both sides) and are quite common.

27 minutes ago, swansont said:

That still relies on curved surfaces

I know.