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The basic elements of the theory of slow invariant manifolds and canard phenomena of singularly perturbed nonlinear differential equations in the context of thermal-explosion problems are outlined. The mathematical results are applied to the investigation of the critical phenomena in autocatalytic combustion models described by singularly perturbed differential equations with lumped and distributed parameters. Critical regimes are modeled by canards (one-dimensional stable-unstable slow invariant manifolds). The geometric approach in combination with asymptotic and numeric methods permits to explain the strong parametric sensitivity and to obtain asymptotic representations of the critical conditions of self-ignition.