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We propose two coding schemes for the two-receiver discrete memorylessbroadcast channel (BC) with rate-limited feedback from one or both receivers.They improve over the nofeedback capacity region for a large class of channels,including the class of \emph{strictly essentially less-noisy BCs} that weintroduce in this article. Examples of strictly essentially less-noisy BCs arethe binary symmetric BC (BSBC) or the binary erasure BC (BEBC) with unequalcross-over or erasure probabilities at the two receivers. When the feedbackrates are sufficiently large, our schemes recover all previously known capacityresults for discrete memoryless BCs with feedback.  In both our schemes, we let the receivers feed back quantization messagesabout their receive signals. In the first scheme, the transmitter simply\emph{relays} the quantization information obtained from Receiver 1 to Receiver2, and vice versa. This provides each receiver with a second observation of theinput signal and can thus improve its decoding performance unless the BC isphysically degraded. Moreover, each receiver uses its knowledge of thequantization message describing its own outputs so as to attain the sameperformance as if this message had not been transmitted at all.  In our second scheme the transmitter first \emph{reconstructs and processes}the quantized output signals, and then sends the outcome as a common updateinformation to both receivers. A special case of our second scheme applies alsoto memoryless BCs without feedback but with strictly-causal state-informationat the transmitter and causal state-information at the receivers. It recoversall previous achievable regions also for this setup with state-information.

Coding Schemes with Rate-Limited Feedback that Improve over the Nofeedback Capacity for a Large Class of Broadcast Channels