Tip! Always shut down ArcMap and do this work in ArcCatalog.

• Vector data
  • Find the correct tool in ArcToolbox
    • Data Management Tools > Projections and Transformations > Feature > Project
  • Add your input Feature Class
  • If your data doesn’t have a defined projection, you will get an error message.
  • Click the Input Coordinate System icon and select the data’s current coordinate system
    • 
  • One easy way to find it is to type the EPSG code in the search text box
    • SWEREF99 (EPSG:3006)
    • RT90 2.5 gon V (EPSG:3021)
    • WGS84 (EPSG:4326)
    • 
    • Click OK
    • Enter the name of the new feature class
    • Select the output coordinate system the same way you selected the current.
    • Select the geographic transformation you want to use from the drop down list.
    • 
    • Click OK

• Raster data
  • Reprojecting raster data is almost the same procedure as for vector data. The only difference is that you can select resampling technique, change cell size, and specify a registration point to align your raster cells with existing rasters.
  • Locate the tool in ArcToolbox 
    • Data Management Tools > Projections and Transformations > Raster > Project Raster
  • 
  • Click OK

Note that the projections SWEREF99 and RT90 are not available for the ESRI GRID format. In order to change the projection you have to convert the raster to a different format (img, tiff,). Use the tool Raster To Other Format in the Conversion Tools > To Raster toolset.