If you liked
Re:Zero kara Hajimeru Isekai Seikatsu
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...then you might like
Summertime Render
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So far, Summertime Render is looking to be one of the best anime shows similar to Re:Zero. Both main characters share an ability "Return by Death" which essentially means returning back in time after death. That said, they have a similar goal, saving the people from a terrible fate that is about to occur. Re:Zero is an Isekai show (the main character is reincarnated into a fantasy world) and Summertime Render is more on the supernatural and mystery side.
If you liked
Higurashi no Naku Koro ni Kai
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...then you might like
Steins;Gate
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Both are incredible stories that flow quite similar one to another. The main characters share the power to go back in time (in Steins;Gate via Sci-Fi, in Higurashi via Fantasy) and are determined to defy fate/destiny. Both present unforgettable and unexpected plot twists that makes you think of any possible future. Higurashi is a lot more violent and gory with scenes that can be hard to watch while Steins;Gate is more of a easy watch kinda.
If you liked
Elfen Lied
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...then you might like
Devilman: Crybaby
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Both have main and side characters with super powers. Both are very bloody, gory and violent with stories that are quite similar (they make you question about the acceptance of any special kind that can be destructive to humanity). In general, I recommend more Devilman Crybaby because it spread that message the best way and also has better animation, soundtrack and characters than Elfen Lied (not saying it’s a bad anime)
If you liked
Shigatsu wa Kimi no Uso
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...then you might like
3-gatsu no Lion
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Both are beautifully written stories with insane character development, top notch animation and soundtrack. The main characters reveal very similar paths in life (both have potencially amazing careers, have suffered a lot and meet someone or people who can change their life).