We study the number of solutions of diophantine equations f(x₁,...,x_n)=v when the variables x_i are restricted to primes. It has been established by Birch and Schmidt that one has the expected number of integer solutions if f is a homogeneous integral polymomial of sufficiently large rank with respect to its degree. We show that the same phenomenon holds when the variables are restricted to primes, extending the results of Hua for diagonal forms. We illustrate some of the ideas on quadratic forms and discuss some elements of the proof of the general case.