<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We study the capacity of multicarrier transmission through a slow frequency-selective fading channel with limited feedback, which specifies channel state information. Our results are asymptotic in the number of subchannels <emphasis><formula formulatype="inline"><tex>$N$</tex></formula></emphasis>. We first assume independent and identically distributed (i.i.d.) subchannel gains, and show that, for a large class of fading distributions, a uniform power distribution over an optimized subset of subchannels, or on–off power allocation, gives the same asymptotic growth in capacity as optimal water filling, e.g., <emphasis><formula formulatype="inline"><tex>$O(\log N)$</tex></formula></emphasis> with Rayleigh fading. Furthermore, the <emphasis><formula formulatype="inline"><tex>$O(\log N)$</tex></formula></emphasis> growth in data rate can be achieved with a feedback rate as <emphasis><formula formulatype="inline"><tex>$O(\log ^{3} N)$</tex></formula></emphasis>. If the number of active subchannels is bounded, the capacity grows only as <emphasis><formula formulatype="inline"><tex>$O(\log \log N)$</tex></formula></emphasis> with the feedback rate of <emphasis><formula formulatype="inline"><tex>$O(\log N)$</tex></formula></emphasis>. We then consider correlated subchannels modeled as a Markov process, and study the savings in feedback. Assuming a fixed ratio of coherence bandwidth to the total bandwidth, the ratio between minimum feedback rates with correlated and i.i.d. subchannels converges to zero with <emphasis><formula formulatype="inline"> <tex>$N$</tex></formula></emphasis>, e.g., as <emphasis><formula formulatype="inline"> <tex>$O\left (\sqrt {{ \log N}\over { N}}\right)$</tex></formula></emphasis> for Rayleigh-fading subchannels satisfying a first-order autoregressive process. We also show that adaptive modulation, or rate control schemes, in which the rate on each subchannel is selected from a quantized set, achieves the same asymptotic growth rates in capacity and required feedback. Finally, our results are extended to cellular uplink and downlink channel models. </para>