study of algebraic equations is one of the oldest and most celebrated themes in mathematics. it was understood that finite systems of equations are decidable in the fields of complex and real numbers. the celebrated hilbert tenth problem stated in 1900 asks for a procedure which, in a finite number of steps, can determine whether a polynomial equation (in several variables) with integer coefficients has or does not have integer solutions. in 1970 matiyasevich, following the work of davis, putnam and robinson, solved this problem in the negative. similar questions can be asked for arbitrary rings. we will give a survey of results on the diophantine problem in different rings and prove the undecidability of equations (the undecidability of the diophantine problem) for free lie algebras of rank at least 3 over an arbitrary field. these are joint results with a. miasnikov.