  We summarize the mathematical basis and practical hints for the explicit
analytical computation of spectral sums that involve the eigenvalues of the
Laplace operator in simple domains. Such spectral sums appear as spectral
expansions of heat kernels, survival probabilities, first-passage time
densities, and reaction rates in many diffusion-oriented applications. As the
eigenvalues are determined by zeros of an appropriate linear combination of a
Bessel function and its derivative, there are powerful analytical tools for
computing such spectral sums. We discuss three main strategies: representations
of meromorphic functions as sums of partial fractions, Fourier-Bessel and Dini
series, and direct evaluation of the Laplace-transformed heat kernels. The
major emphasis is put on a pedagogic introduction, the practical aspects of
these strategies, their advantages and limitations. The review summarizes many
summation formulas for spectral sums that are dispersed in the literature.
