The book is devoted to a classical subject: solving linear PDEs by means of Laplace, Fourier and the lesser known Hankel transforms. Preliminary knowledge of complex integration (including multi-valued functions) and Bessel functions is assumed, though a short introduction is given. The core of the book consists of several hundred mostly worked solutions of PDEs, typically on two-dimensional (semi)infinite domains. Both Cartesian and polar coordinates and various combinations of boundary conditions are present. Each section is devoted to a particular transform or a combination of these. The last chapter deals with a more advanced Wiener-Hopf technique. An extensive bibliography of papers using each particular method is given. Special attention is paid to computing inverse transforms and various analytic and numerical techniques are discussed. The discussed solutions typically involve quite difficult algebra, while non-trivial mathematical steps, such as a change of order of integration or expansion into infinite series (product) are not justified. This gives the book more of an 'engineering' flavour; nonetheless it is certainly of interest to a wider audience.