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.