  We derive explicit upper bounds for various functions counting primes in
arithmetic progressions. By way of example, if $q$ and $a$ are integers with
$\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of
the logarithms of the primes $p \equiv a \pmod{q}$ with $p \leq x$, we show
that $$ \bigg| \theta (x; q, a) - \frac{x}{\phi (q)} \bigg| < \frac1{160}
\frac{x}{\log x}, $$ for all $x \ge 8 \cdot 10^9$ (with sharper constants
obtained for individual such moduli $q$). We establish inequalities of the same
shape for the other standard prime-counting functions $\pi(x;q,a)$ and
$\psi(x;q,a)$, as well as inequalities for the $n$th prime congruent to $a\pmod
q$ when $q\le1200$. For moduli $q>10^5$, we find even stronger explicit
inequalities, but only for much larger values of $x$. Along the way, we also
derive an improved explicit lower bound for $L(1,\chi)$ for quadratic
characters $\chi$, and an improved explicit upper bound for exceptional zeros.
