This paper considers the pricing of European Asian options in the Black-Scholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. Firstly we consider the partial differential equation approach to the pricing of Asian options. We show the link between the approaches of Rogers and Shi [1995], Andreasen [1999], Hoogland and Neumann [2000] and Vecer [2001]. For the latter two formulations we propose two reductions, which increase the numerical stability and reduce the calculation time.Secondly, we show how a closed-form expression can be derived for Rogers and Shi's lower bound for the general case of multiple underlyings. Thirdly, we sharpen Thompson's [1999a,b] upper bound for the value of an Asian option. This is important for the practically relevant case of options with long maturities. Numerical results show that when the strike price is not extremely high, the resulting upper bound is tighter than recently introduced upper bounds in studies by Nielsen and Sandmann [2003] and Vanmaele et al. [2005].Finally, we consider analytical approximations for the value of an Asian option. A much heard criticism on moment-matching approaches is that the error in the approximation is not known beforehand. We combine the traditional moment-matching approaches (e.g. Levy [1992]) with the conditioning approaches (e.g. Curran [1994]) and introduce a class of analytical approximations, which can be proven to lie between a sharp lower and upper bound. In numerical examples the accuracy of these new approximations is demonstrated. The approximations are found to outperform all of the current state-of-the-art upper bounds and approximations.