It's not very well-known, mostly by local residents. Therefore, it's not swarming with people. The view is great and when the pagan church is open on weekends you might be able to climb on the roof for even better view.
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Everyone knows the viewpoints from Gediminas' Castle Tower Hill and Hill of Three Crosses, but Bekesh Hill is yet another good spot to stand above the city and admire the view. You can go there as sequence to your Užupis sightseeing walk. There are two other possible paths to it but I would like to share the one I discovered and know.
Starting at the angel statue keep to the road on your left and walk for 10 mins until you see steps on your left, near a private house. Climb them.
Mon-Su: Open 24 hours