It is a classical problem of algebraic geometry to decide whether or not the field of moduli of an algebraic curve is a field of definition. The first results go back to Weil and Shimura, however  progress has been slow since. After a brief introduction of the problem we focus on the hyperelliptic curves with extra automorphisms. We conjecture that for such curves the field of moduli is a field of definition. Using a new set of invariants we prove this conjecture for curves with three involutions and several other families. Our approach is constructive and in all cases finds a rational model of the curve over its field of moduli.