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Cryptocurrencies have recently emerged as a popular asset class, with investors having high risk appetite and speculative attributes. They are not backed by physical assets, such as commodities or real currencies; they are purely speculative assets having high volatility. Regulatory authorities across the globe have conflicting rules regarding cryptocurrencies. Recent studies on volatility of cryptocurrencies have primarily addressed univariate volatility analysis and volatility spillover between cryptocurrencies and other asset classes, mostly stocks and commodities. This study has three objectives. Firstly, it considers six prominent cryptocurrencies, i.e., Bitcoin, Ethereum, Binance Coin, Cardano, Tether, and Ripple, and examines the nature of asymmetrical volatility in them using EGARCH and TGARCH techniques. Secondly, it examines whether there are volatility spillovers between the cryptocurrencies as well as from one of the most popular global fear indices, i.e., CBOE volatility index, using dynamic conditional correlation (DCC). Thirdly, it further measures the total and directional volatility spillover among the cryptocurrencies using the Diebold-Yilmaz index. This study has found that Ethereum and Ripple may be used to construct a portfolio. There exists long-term volatility spillover among all the cryptocurrencies; however, there is no short-term spillover of volatility. Volatility of Binance Coin, Cardano, and Ripple influence and are influenced the most by volatilities of other cryptocurrencies.