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Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence\/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures \u2013 parametrized probability measures which can describe the limits of such oscillatory sequences \u2013 offer the more general paradigm of<jats:italic>measure-valued solutions<\/jats:italic>for these problems.<\/jats:p><jats:p>We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. 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