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Light absorption by a spatially uniform non-magnetic spherical nanopartile in the vicinity of surface plasmon (polariton) resonances is studied in detail based on the exact Mie solution. It is shown that, in sharp contrast to the common belief, a weakly dissipating particle absorbs much more energy than a particle with strong losses. The absorption may have very unusual properties and cannot be described by the Rayleigh approximation, no matter how small the particle is. A simple universal formula for the resonant absorption lineshape as a function of the particle size and its complex dielectric permittivity is obtained. The obtained results are applied to the problem of laser heating of nanoparticles. The optimization of the input of the laser pulse energy into a particle is performed at fixed fluence with respect to the particle size, wavelength and duration of the laser pulse. We introduce a new quantity, an effectiveabsorption coefficient of a particle, which allows to compare quantitatively the light absorption by nanoparticles with the absorption of a bulk material. We describe the range of parameters where giant absorption enhancement can be observed, and give practical examples for metals whose optical properties vary from weak (potassium) to strong (platinum) dissipation. We also point out that even larger absorption can be achieved for core-shell nanoparticles. Effects of heat transfer from the particle to a host medium are inspected too. The results obtained may be employed in biology, medicine, storage and processing of information and in other nanotechnologies.