Smooth\, multivariate funct ions defined on tensor domains can be approximated using orthonormal bases formed as tensor products of one-dimensional orthogonal polynomials. On th e other hand\, constructing orthogonal polynomials in irregular domains is difficult and computation ally intensive. Yet irregular domains arise in ma ny applications\, including uncertainty quantifica tion\, model-order reduction\, optimal control and numerical PDEs. In this talk I will introduce a framework for approximating smooth\, multivariate functions on irregular domains\, known as polynomi al frame approximation. Importantly\, this approa ch corresponds to approximation in a frame\, rathe r than a basis\; a fact which leads to several key differences\, both theoretical and numerical in n ature. However\, this approach requires no orthog onalization or parametrization of the domain bound ary\, thus making it suitable for very general dom ains\, including a priori unknown domains. I will discuss theoretical result s for the approximatio n error\, stability and sample complexity of this approach\, and show its suitability for high-dimen sional approximation through independence (or weak dependence) of the guarantees on the ambient dime nsion d. I will also present several numerical re sults\, and highlight some open problems and chall enges. LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR