Abstract: Algebraic topology studies algebraic invariants of topological spaces, typically identifying spaces which are "homotopy equivalent." Homotopy theory proved essential in the study of topological spaces; when applied to algebraic varieties, its ideas turn out to be even more powerful, as the resulting invariants have far more structure, often coming from either complex analysis (Hodge theory) or arithmetic (an action of a Galois group). In particular, as predicted e.g. by the Hodge conjecture, the Tate conjecture, or Grothendieck?s section conjecture, they should be able to detect algebraic cycles and rational points. Over the field of complex numbers, an algebraic variety gives rise to a nice topological space to which one can apply methods of algebraic topology directly. From the point of view of arithmetic and algebraic geometry over more general fields, it is of key importance to construct algebraically defined topological invariants. Probably the most successful development to this date achieving this goal is the introduction of the etale topology by Artin and Grothendieck, and the etale homotopy type of Artin and Mazur. I will give a gentle introduction to this circle of ideas, explain several ways in which one can associate a homotopy type to an algebraic variety, and discuss how the resulting objects witness phenomena quite foreign to classical algebraic topology of nice topological spaces.