There have been many studies of transport in an unbounded fractal medium. Here we discuss a number of quantities related to the concentration of reactants diffusing in fractal media in the presence of a trapping point. This investigation is suggested by an extension of the Smoluchowski model to calculate reaction rates in such media [M. von Smoluchowski, Phys. Z. 16, 321 (1915); 17, 557 (1917); 17, 585 (1917)]. Results, some analytic and some based on a scaling argument, are given for the flux into the trap, which is the analog of the reaction rate, in addition to the concentration profile in the neighborhood of the trap and the time dependence of the distance between the trap and the nearest untrapped particle. The results of our theory are found to be in good agreement with Monte Carlo and exact enumeration calculations for the concentration profile on a Sierpinski gasket and on an infinite percolation cluster at criticality. Some scaling and numerical results are reported for the situation in which the traps move in the presence of fixed particles.