Fundamental Approximation Methods. Machine Learning -- Deep Neural Nets -- Chebyshev Tensors -- The toolkit - plugging in approximation methods. Introduction: why is a toolkit needed -- Composition techniques -- Tensors in TT format and Tensor Extension Algorithms -- Sliding Technique -- The Jacobian projection technique -- Hybrid solutions - approximation methods and the toolkit. Introduction -- The Toolkit and Deep Neural Nets -- The Toolkit and Chebyshev Tensors -- Hybrid Deep Neural Nets and Chebyshev Tensors Frameworks -- Applications. The aim -- When to use Chebyshev Tensors and when to use Deep Neural Nets -- Counterparty credit risk -- Market Risk -- Dynamic sensitivities -- Pricing model calibration -- Approximation of the implied volatility function -- Optimisation Problems -- Pricing Cloning -- XVA sensitivities -- Sensitivities of exotic derivatives -- Software libraries relevant to the book -- Appendices. Families of orthogonal polynomials -- Exponential convergence of Chebyshev Tensors -- Chebyshev Splines on functions with no singularity points -- Computational savings details for CCR -- Computational savings details for dynamic sensitivities -- Dynamic sensitivities on the market space -- Dynamic sensitivities and IM via Jacobian Projection technique -- MVA optimisation - further computational enhancement.