The paper presents the robust estimates of tail indices for financial returns and returns asymmetry in the Russian stock market. We also investigate the relation between individual characteristics of companies and the degree of heavy-tailedness and asymmetry of returns. According to our estimates, the degree of heavy-tailedness is strongly related to the liquidity of stocks and the company size. At the same time, no significant effects on estimates of the tail indices of sectoral affiliation, cross-listing, adding into quotation lists, state ownership are revealed. As for the influence of the above-mentioned factors on the asymmetry of returns, the statistical reliability of relevant models is rather low. However, certain indicators of the asymmetry are observed for medium-sized regional companies, the majority showing a heavier right tail compared to the left tail. We also discuss the implications of our findings for managerial decisions and economic modeling. Our results may be useful for risk-managers, financial regulators, investors and policy-makers.