We are interested in the quantity $\rho$(q, g) defined as the smallest positive integer such that r $\ge$ $\rho$(q, g) implies that any absolutely irreducible smooth projective algebraic curve defined over F q of genus g has a closed point of degree r. We provide general upper bounds for this number and its exact value for g = 1, 2 and 3. We also improve the known upper bounds on the number of closed points of degree 2 on a curve.