The aim of the talk is to introduce the notion and present results on ring endomorphisms having large images.
In particular, we will show that:

(i) an endomorphism $\sigma$ of a prime one-sided noetherian ring $R$ is injective iff the image $\sigma(R)$ contains an ideal $I$ of $R$. If additionally $\sigma(I)=I$, then $\sigma$ has to be an automorphism of $R$.

(ii) The Jacobian conjecture has a positive solution for endomorphism with large images.

Examples showing that the assumptions imposed on $R$ can not be weakened to $R$ being a prime Goldie ring will be presented and some problems will be formulated.