On the scale of most astronomical phenomena, general relativity (GR) is the relevant theory.
There are many aspects in which this theory is incompatible with Euclidean geometry. An illustrative and often used analogy is that Euclidean geometry is already inaccurate for a being confined to the surface of a sphere - if you draw a triangle on a sphere, its interior angles do not in general sum up to 180°. What makes GR a bit strange still is that it posits that it is not space alone that participates in such curved geometry, but spacetime.
That is, general relativity does not assume that space(time) is flat, and it even intermingles space and time so that different observers that move relative to each other will neither agree on whether two arbitrary events are synchronous nor whether they happen at the same place.
The specific physical effects this has are too varied to discuss them here at length and have already been extensively discussed on this site, see e.g. How can time dilation be symmetric?, What is the proper way to explain the twin paradox? and many other questions in the general-relativity tag.