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I am very new to QGIS and GIS in general, so I am struggling with the following: when I do a map of Russia, I get the part next to Alaska being drawn far left from Europe, as it lies behind 180o... I could not find any CRS that depicts Russia "in full"...

Can someone please explain to me how wrap the map in being "one chunk", not being split by the 180 longtitude?

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You can follow my tutorial here: QGIS display world country shape files centered on pacific ocean using Robinson, Miller Cylindrical or other projection

Using 100°E as central meridian, you have to cut the world at -80°E, and you will get Russia in full. It does not have to be Robinson or Miller projection, any other world wide projection should work:

Miller:
+proj=mill +lat_0=0 +lon_0=100 +x_0=0 +y_0=0 +R_A +a=6371000 +b=6371000 +units=m +no_defs

Miller 100°E

Mercator:
+proj=merc +lon_0=100 +k=1 +x_0=0 +y_0=0 +a=6371000 +b=6371000 +units=m +no_defs

Mercator 100°E

Robinson:
+proj=robin +lon_0=100 +x_0=0 +y_0=0 +a=6371000 +b=6371000 +units=m +no_defs

Robinson 100°E

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