In this paper we establish new upper and lower bounds for the nth prime number p_n, which improve several existing bounds of similar shape. As the main tool, we use some explicit estimates recently obtained for the prime counting function. A further main tool is the use of estimates concerning the reciprocal of log p_n. As an application, we derive new estimates for ϑ(p_n), where ϑ(x) is Chebyshev's ϑ-function.