I have a shapefile with contours and their heights.
Is it possible to make a DEM from contours in QGIS without using GRASS?
I found Creating DEM from contours using ArcGIS Desktop? but the answer is for ArcGIS Desktop.
Yes, there are several options available in QGIS:
Raster > Grid (Interpolation). GDAL's interpolation is more robust because you can use other interpolation algorithms (IDW, nearest neighbor, moving average, etc.). This tools only works for point data.
r.surf.*). You need convert your shapefile into a GRASS database to use the GRASS modules in QGIS.
GRASS has also many options
available one way or another also trough the QGIS/GRASS plugin.
The question is old, but as it is ranked high in Google search, so I think it is helpful to say QGIS has built-in interpolation tools:
However, the tutorial is also old.
For QGis 3.2, you can find it in the PROCESSING TOOLBOX, or in the RASTER > ANALYSIS > GRID menu: