  This survey summarizes briefly results obtained recently in the Casimir
energy studies devoted to the following subjects: i) account of the material
characteristics of the media in calculations of the vacuum energy (for example,
Casimir energy of a dilute dielectric ball); ii) application of the spectral
geometry methods for investigating the vacuum energy of quantized fields with
the goal to gain some insight, specifically, in the geometrical origin of the
divergences that enter the vacuum energy and to develop the relevant
renormalization procedure; iii) a universal method for calculating the high
temperature dependence of the Casimir energy in terms of heat kernel
coefficients. A special attention is payed to the mathematical tools applied in
this field, namely, to the spectral zeta function method and heat kernel
technique.
