What is Risk contribution, risk budgeting, equal risk contribution portfolio, risk parity?

The risk contribution of position $i$ is $w_i(\Sigma w)_i/(w^\top\Sigma w)$, its Euler share of the variance (Book 6, chapter 20); the shares sum to one. Risk budgeting chooses long-only weights whose risk contributions equal given budgets; the equal risk contribution portfolio gives every position the same share (Maillard, Roncalli and Teïletche). Risk parity is the practice of allocating across asset classes by equal risk contributions, usually with leverage.