The decision-making situation under risk is defined and the certainty equivalent of a lottery with uni-dimensional outcomes is introduced. An agent is globally risk averse if the risk premium (defined as the difference between the expected value and the certainty equivalent) of every lottery is nonnegative. This definition is used in order to show some characteristics of the certainty equivalent function that are associated with risk aversion. Risk aversion is equivalent to concavity of utility function if the expected utility theory holds. The comparison of risk aversion across agents is examined too. The notion of local risk aversion is introduced in general and with respect to the expected utility case, where again it is equivalent to concavity of utility function. A measure of risk aversion, if expected utility theory holds, is provided by the de Finetti-Arrow-Pratt index. Aversion towards increases in risk, i.e. preference for less risky lotteries, is then examined with respect to two ways that increase risk: mean preserving spreads and probability mixtures. The equivalence of the former with second-order stochastic dominance is examined. Finally, risk aversion and aversion to increasing risk with reference to the rank dependent expected utility model is examined mainly in order to show a kind of risk aversion (first-order risk aversion) which is stronger than the risk aversion permitted by the expected utility theory (second-order risk aversion).