We present first results of spectroscopic observations of transiting exoplanets in the Special Astrophysical Observatory of the Russian Academy of Sciences with the Main Stellar Spectrograph of the 6-m BTA telescope. For the exoplanetWASP-32b, we detected a significant variation of intensity and equivalent width in the H alpha spectral line of the parent star at the time of a transit. The equivalent width of the line during transit is by 8-10% larger than outside the planet passage. Residual intensity in the core of the line reveals the following tendency: the line is by 10-15% deeper inside transit than outside it. Observations with the long-slit spectrograph of the Crimean Astrophysical Observatory at the 2.6-m ZTSh telescope also showed a transit event in the H alpha line, although, with a smaller amplitude and shape inverted in relation to the data from the 6-m telescope. While in the observations with the BTA the H alpha line becomes deeper during the transit, in the ZTSh observations, the residual intensity of the H alpha line decreases during the transit. Reducing and analysis of the archive data of WASP-32b observations with the HARPS spectrograph also confirm the H alpha line modulation at the time of the transit. The observed data give evidence of the envelope in WASP-32b filling the Roche lobe and a comet-like tail of changing geometry and orientation relative to the observer. These changes determine different depths and shapes of the H alpha spectral line at the time of transits.