I dowloaded the climate data from this site http://catalogue.ceda.ac.uk/uuid/3f8944800cc48e1cbc29a5ee12d8542d and they have the extension .dat . I'd like to import the information in QGIS as raster data.

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    What do you see if you open the .dat file with text editor? – user30184 Aug 10 '16 at 12:15
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    From the menubar, go to: Layer > Add Layer > Add Raster Layer... and try selecting the ZMap Plus Grid (*.dat *.DAT) extension. – Joseph Aug 10 '16 at 12:16

From the More Information page:

The CRU TS 3.00 data (i.e. climate variables) are available in both compressed (.gz extension) ASCII ".dat" and netcdf ".nc" file formats. CRU TS 3.00 metadata files (i.e. stations) are available in ASCII. To understand how to read the CRU TS 3.00 data and metadata files, please refer to the CRU_TS_3.00_File_Formats_explained PDF documentation (under Linked Documentation below).

So you have an ASCII file - that means you can open it with a text editor or on unix do something like head file.dat to see what it looks like. It may well be

0 0 139392.2
0 1 3082792.2

Which is an XYZ file or it might be:

ncols         4
nrows         6
xllcorner     0.0
yllcorner     0.0
cellsize      50.0
NODATA_value  -9999

which is an ESRI style Ascii Grid.

QGIS can open either of these and many other variations.

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