  We propose two coding schemes for the two-receiver discrete memoryless
broadcast 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 we
introduce in this article. Examples of strictly essentially less-noisy BCs are
the binary symmetric BC (BSBC) or the binary erasure BC (BEBC) with unequal
cross-over or erasure probabilities at the two receivers. When the feedback
rates are sufficiently large, our schemes recover all previously known capacity
results for discrete memoryless BCs with feedback.
  In both our schemes, we let the receivers feed back quantization messages
about their receive signals. In the first scheme, the transmitter simply
\emph{relays} the quantization information obtained from Receiver 1 to Receiver
2, and vice versa. This provides each receiver with a second observation of the
input signal and can thus improve its decoding performance unless the BC is
physically degraded. Moreover, each receiver uses its knowledge of the
quantization message describing its own outputs so as to attain the same
performance 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 update
information to both receivers. A special case of our second scheme applies also
to memoryless BCs without feedback but with strictly-causal state-information
at the transmitter and causal state-information at the receivers. It recovers
all previous achievable regions also for this setup with state-information.
