In this Thesis, two problems were studied: a direct vacuum acceleration of electrons by a tightly focused ultrashort relativistic laser pulse and ion acceleration in the process of spherical laser-heated plasma explosion. The electromagnetic field of a tightly focused laser pulse was evaluated numerically by means of Stratton-Chu integrals. The properties of the focused field were analyzed in detail for a plane wave or a macroscopically large Gaussian beam incident onto the mirror. Free electrons moving in the tightly focused field were found to accelerate by two possible mechanisms: focal spot acceleration and capture-and-acceleration scenario. The two mechanisms were studied in detail. Comparison of the mirror-focused field with first- and fifth-order paraxial fields is performed. A 3D electromagnetic PIC code SCPIC was created for simulations of pulse interaction with targets having a finite number of particles interacting with each other by collective fields. Atto-second bunch formation was observed in the interaction with ultra-small or ultra-thin targets. Physical mechanism of bunch formation is explained. The problem of electrostatic explosion of a nano-scale spherical plasma with initially hot electrons and cold ions was solved numerically. Expansion in a wide regime of electron temperature $0 < T leq infty$ was studied in detail for different initial density profiles of plasma. Favorable conditions for obtaining mono-energetic ions resulting from the explosion were specified in single and two ionic species cases. In case of a two-species explosion, the number of mono-energetic, $deltavarepsilon/varepsilon < 10\%$, ions can be as high as 70-80\% of the total light ions for a wide range of electron temperatures.

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