In this talk, I will try to motivate the equations of elasto-plasticity as the simplest departure from the
fully reversible setting of elasticity. I will then discuss the current mathematical state of that field and focus on
issues of uniqueness which suggest a strong parallel with similar questions in fluid dynamics. I will conclude with an incursion into
the vagaries of finite elasto-plasticity which remains, in spite of its technological relevance, a complete mystery.