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The unifying theme of this series of talks is the classical problem of counting lines in the projective models of K3-surfaces of small degree. Starting with such classical results as Schur's quartic and Segre's bound (proved by Rams and Schütt) of 64 lines in a nonsingular quartic, I will discuss briefly our recent contribution (with I. Itenberg and A. S. Sertöz), i.e., the complete classification of nonsingular quartics with many lines. There are limitless opportunities in extending and generalizing these results. First, one can switch from the complex numbers to an algebraically closed field of characteristic p>0. Here, of course, most interesting are the so-called (Shioda) supersingular surfaces. I will discuss the properties of (quasi-)elliptic pencils on such surfaces, culminating in the classification of large configurations of lines for p=2,3. Alternatively, one may consider non-closed fields such as R or Q. For the former, the sharp bound is 56 real lines in a real quartic; for the latter, the current bound is 52, and the best known example has 46 lines.