In the 1960's Richard Thompson defined three groups nowadays denoted by T, F, and V that have remarkable properties, e.g. they are finitely presented but have unsolvable word problem, T and V are simple, F has property FP∞ and is torsion-free. All three groups, as well as numerous generalisations, are still subject to much research. Our plan is to follow the Introductory Notes on Richard Thompson's Groups  by Cannon, Floyd, and Parry, and afterwards to look at a selection of more recent results.
Word Growth in Groups
Bass-Serre Theory and Profinite Analogues
p-Adic analytic pro-p groups
Invariant random subgroups
Probabilistic methods in group theory