ABSTRACT: An introduction to fractional derivatives and some of their properties will be presented. The regularity of solutions to Caputo fractional initial-value problems is then discussed; it is shown that typical solutions have a weak singularity at the initial time $t=0$. This singularity has to be taken into account when designing and analysing numerical methods for the solution of such problems. To address this difficulty we use graded meshes, which cluster mesh points near $t=0$, and answer the question: how exactly should the mesh grading be chosen? Finally, initial-boundary value problems are considered, where the time derivative is a Caputo fractional derivative. (This is a fractional-derivative generalisation of the classical parabolic heat equation.) Once again a weak singularity appears at $t=0$, and the mesh in the time coordinate should be graded to compute satisfactory numerical solutions. This problem is the most widely studied fractional-derivative problem in the numerical analysis literature and a brief outline will be given of this work. BIO: Martin Stynes received his MS in 1972 from University College Cork, Ireland and Ph.D. in Mathematics, Oregon State University. Martin Stynes was a Lecturer at Waterford Regional Technical College, Ireland from 1979 to 1984. He then moved to University College Cork, where he was successively Lecturer, Statutory Lecturer, Professor. He was a Fulbright Research Professor at Kent State University (USA) in 1990, Royal Society Visiting Fellow at Oxford University (U.K.) in 1991, Visiting Professor at University of Colorado at Denver (USA) in 1997, DFG Visiting Professor at Otto von Guericke University Magdeburg (Germany) in 1998, Visiting Professor at the University of South Carolina (USA) in 2004, and DFG Mercator Guest Professor at the Technical University of Dresden (Germany) in 2008. He was the elected President of the UK and Republic of Ireland Section of SIAM 2003-5. He serves on the editorial boards of the journals Advances in Computational Mathematics, Computational Methods in Applied Mathematics, Fractional Calculus and Applied Analysis, Journal of Scientific Computing, and Mathematical Proceedings of the Royal Irish Academy. He took early retirement from University College Cork in 2012 in order to concentrate on research. He has been at CSRC since 2014.