#### DOCUMENTA MATHEMATICA, Extra Volume ICM I (1998), 579-604

Title: ${\bf A}^1$-Homotopy Theory
${\bf A}^1$-homotopy theory is the homotopy theory for algebraic varieties and schemes which uses the affine line as a replacement for the unit interval. In the paper I present in detail the basic constructions of the theory following the sequence familiar from standard texbooks on algebraic topology. At the end I define motivic cohomology and algebraic cobordisms and describe algebraic K-theory in terms of this theory.