How does RNA polymerase know where to stop transcribing?

RNA polymerase identifies where to stop transcribing by recognizing specific sequences in the DNA template called termination signals or terminators. These sequences indicate the end of the gene or transcription unit. Once RNA polymerase encounters a termination signal, it halts transcription and releases the newly synthesized RNA molecule. 

In prokaryotes, termination of transcription occurs through two mechanisms: rho-dependent and rho-independent. Rho-dependent termination relies on a protein called Rho factor. When RNA polymerase transcribes certain genes, it encounters a specific DNA sequence called the rho utilization (rut) site. Rho-independent termination, also known as intrinsic termination, relies on the formation of a stable RNA hairpin structure followed by a run of uracil (U) nucleotides in the mRNA transcript. 

In eukaryotes, transcription termination involves two elements: a downstream terminator sequence and a poly(A) signal. Transcription termination in eukaryotes involves the recognition of a polyadenylation signal sequence (AAUAAA) in the pre-mRNA transcript. This sequence signals the addition of a polyadenylate (poly(A)) tail to the mRNA transcript. The cleavage and polyadenylation complex, which includes specific proteins, recognizes the poly(A) signal and cleaves the pre-mRNA transcript downstream of the signal. Sequences, located downstream of the poly(A) signal, contribute to the termination of transcription by promoting the release of RNA polymerase from the DNA template.