This book deals with some of the recent advances of the theory of Backward Stochastic Differential Equations (BSDE) with generators that grow quadratically in the control variable (qgBSDE). One starts by looking at the differentiability of these equations in its several parameters, the relevance of which is clarified as one builds new results for this class of equations, like path regularity or explicit convergence rates for truncated qgBSDE. Both these results are the first of its kind for qgBSDE, as they allow at last the required theoretical justification for the use of numerical methods to approximate the solution qgBSDE. The exponential transformation as a reduction method of qgBSDE to standard Lipschitz BSDE is also discussed. This leads once again to the theme of numerical approximation for this class of BSDE. The book concludes with a problem of optimal investment of insurance related derivatives written on non-tradable underlyings, but correlated with tradable assets. Dynamic utility-based indifference prices are calculated and closed form formulas for the prices and for the derivative hedges are given.