It is fundamental to understand the distribution of primes in arithmetic progressions. With the aid of Brun’s sieve, Titchmarsh gave the first upper bound, which is of correct order of magnitude, on the number of such primes in an individual arithmetic progression. This gives the so-called Brun--Titchmarsh theorem. We will discuss our recent work on sharpening this theorem with better constants for general moduli and for special moduli. If time permits, we also mention its connection with Landau--Siegel zeros and subconvex bounds for Dirichlet L-functions. This is a joint work with Junren Zheng.
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