What are alternative measures of risk? In finance, the variance of the returns of a security are used as a proxy for the associated risk of the security. I've seen some books include sentences like "if you take variance as a measure of risk...". In what way might variance be an incomplete measure of risk? What alternatives do we have?
A variance is an incomplete measure of risk in a sense, that it measures uncertainty in security payoffs, rather than uncertainty in holder's welfare. In the simplest way we can demonstrate this point as follows.

Suppose that agents want to marginally increase her holding of an asset by $\xi$ and a unit of asset provides a payoff of $x$, which is a random variable. Now we model agent as caring about volatility in her consumption $c$, which is more reasonable, than assumption that she cares about volatility in the payoff directly. Then $$Var(c+\xi x) = Var(c) + 2\xi\cdot cov(c,x) + \xi^2Var(x)$$ and the last term is negligible as $\xi$ is marginal change. 

So, as marginal changes are considered in equilibrium, it is more reasonable to take covariance between consumption and payoff as measure of riskiness of the security, rather than just variance of the payoff. In practice the aggregate consumption is often used as a proxy for individual consumption.

Yet another more accurate measure of riskiness can be introduced if we were to presume not direct care for volatility in consumption, but consider an agent with a utility dependent upon his consumption stream. It is a rather lengthy technical exercise and those who are interested should look it up in an excellent textbook "Asset Pricing" by John Cochrane. I will limit myself with presenting the result for multiperiod discrete time model here.

If we were to defined a random variable $m_t = \beta\frac{u'(c_{t+1})}{u'(c_t)}$ called stochastic discount factor for agent with von Neumann-Morgenstern utility and discount factor $\beta$ then by the similar argument as above we may derive that $cov(m_t,x_t)$ should be considered to be even a better measure of riskiness of the security.

Obviously, any nonrandom transformation of the above may be considered to be measure of riskiness, namely, standard deviation and correlation instead of variance and covariance are quite popular. Also different nonrandom normalizations may be applied to $m_t$. For example $u'(c)$ may be used, as $\beta$ is nonrandom and $c_t$ is known at time $t$