The natural sciences are where funded doctorates are most common. Universities in Sweden, Norway, Denmark and Switzerland employ PhD candidates on salaries, and their research groups advertise openly to applicants from any country. Right now our funded positions page carries posts in physics, chemistry, astronomy and bioinformatics.
At master's level the picture splits in two. Germany and Austria offer degrees that are free or nearly free in tuition, at Freie Universität Berlin and the University of Vienna, but you pay your own living costs. Full awards that also cover living costs are rarer and more competitive: ETH's ESOP asks for the top 10% of your cohort, and the University of Helsinki grants its waivers programme by programme.
If you already hold a master's, two routes are unusual enough to plan around. The University of Iceland keeps its doctoral programmes free of tuition, and Ghent University's sandwich PhDs split your doctorate between Belgium and a partner university at home.
Science applications rise and fall on research fit. Before you apply for a doctorate, read your future supervisor's recent papers, and say in your first email which of their questions you want to work on.