The irreversible adsorption of telechelic polymer chains from solution and melts onto solid substrates has been studied using the bond fluctuation Monte Carlo model. Complex brush formation kinetics dominated by diffusion of chains to the substrate at short times (diffusion-limited regime or DLR) and by penetration of chains through the maturing brush at longer times (penetration-limited regime or PLR) were observed. During the entire adsorption process, the rate of chain adsorption decreases monotonically with time. In the DLR, characterized by a maximum in the concentration of singly bound chains and a rapidly increasing fraction of doubly bound chains (loops), this decrease is due primarily to the depletion of free chains near the substrate and the formation of concentration gradients of free (nonadsorbed) chains in the bulk solution. The DLR and PLR are separated by an intermediate regime during which the brush becomes dominated by doubly bound chains and both penetration of the maturing brush and diffusion of chains to the brush surface play a role in determining the kinetics of brush growth. The PLR is characterized by steep gradients of free chains within the growing brush and the disappearance of concentration gradients for free chains in the bulk solution. In the PLR, the concentration of singly bound chains is low and decreases slowly while surface coverage and the fraction of doubly bound chains increase slowly. The rates of adsorption of new chains and the formation of loops in the PLR slow dramatically with increasing surface coverage and increasing chain length and less dramatically with decreasing bulk concentration.