The problems of acceleration of fast ignition thermonuclear targets with two cones for their high-precision injection into region near the center of the reactor chamber are considered and the possibility of solution of these problems is shown. A brief review of discussed variants of such targets and of their main advantages, related to ignition of microexplosion and simplicity of providing preservation of targets workability during their flight in the reactor chamber, is presented. Fast ignition by microexplosion of two-sided cone target and the method to estimate acceptable speed of stabilizing rotation of thermonuclear target are proposed. Keywords: controlled thermonuclear fusion, inertial confinement, fast ignition, two-sided cone targets, spin stabilization of flight.