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We consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x^(α)(t)=f(t,x(t)), x(0)=x₀, where x^(α) is the derivative of x of order α in the sense of Caputo and 0<α<1. We review some of the existing methods and explain their respective strengths and weaknesses. We identify and discuss potential problems in the development of generally applicable schemes.