Mathematical Logic: Logic done with mathematical methods.
Aristotle discovered or invented logic as a science, and indeed also introduced variables in it, in a mathematical way, but so far as mathematics and its methods and notations are concerned, this is were logic remained the next 2000 years or so.
Leibniz was the first - probably inspired by Pascal and Arnauld - to think of logic as a mathematical subject, but he did not publish his work.
Mathematical logic was created in the 19th Century, by George Boole - see his The Mathematical Analysis of the Laws of Thought - and then further developed by Jevons, Peirce, and Frege. Frege also seems to have been the first logicist i.e. someone who held that mathematics in fact can be derived from and founded on logic.
Frege's work, and also Peirce's, was hardly reviewed or known during their lifetimes, and the first widely known work in mathematical logic was the Principia Mathematica, by Whitehead and Russell, that attempted - in three large volumes, mostly filled with mathematical notation, to provide a foundation for both mathematics and logic, and to derive mathematics from logic.
The Principia Mathematica was a failure in that Whitehead and Russell found that they could not do without axioms - such as the axioms of infinity and choice - that were not logical in any clear intuitive sense and provide a foundation for mathematics, and also required assumptions - as are involved in the theory of types - that seem neither necessary nor mathematical, but were introduced by Whitehead and Russell to prevent the arisal of paradoxes, such as Russell's Paradox, that exploded Frege's system of logic and mathematics.
Since then mathematical logic has been rapidly developed, mostly by mathematicians, and some philosophers also. An important text was Hilbert and Bernays "Grundlagen der Mathematik", and important theorems of the twenties and thirties were those of Skolem and Gödel (completeness and incompleteness theorems) whereas Church, Turing, and Tarski articulated important theories and concepts (lambda-abstraction, effective computability, and truth and model theory).
The work of Gödel (incompleteness theorems) refuted the attempts of Hilbert and Bernays to give mathematics a proof-theoretical foundation, in which every true mathematical proposition would get its proof.
The work of Church, Turing and Tarski sparked off a great amount of work, much of it quite mathematical and beyond non-specialists, in the second half of the 20th C, and gave rise to electrical programmable computers (Turing, Von Neumann), to programming languages (such as Lisp, founded on lambda-abstraction) and many subtle results about proofs and models, that refined Gödel's and Tarski's ideas (see Smullyan).
Another logical subject that was mathematiced in the second half of the 20th C was modal logic, especially after Kripke showed how this could be done using model theoretical tools.
Also, having mentioned logicism, three schools of thought arose about the foundations of mathematics in the 20th Century, namely the logicists - Russell, Carnap - who believed mathematics could be founded on logic; the intuitionists - Brouwer, Weyl, Heyting - who denied it could, and who also denied the logical validity of what where up to then accepted logical axioms (viz. (~~p --> p) and (~p V p)), in the end on the ground that one could only regarded as proved what one could show effectively (constructively) how to construe; and the formalists (Curry, Feferman, Boolos) who considered both mathematics and logic as precise formalized kinds of reasoning with symbols, only constrained by the need for consistency.
There were several important foundational results in the 20th Century, such as the proof that the Axiom of Choice is independent of the other axioms of standard set theory (Gödel, Cohen); the discovery of the non-standard reals as an alternative foundation for the calculus (Robinson); the discovery of tableaux as a system of proofs (Beth, Smullyan); the refinement of Gödel's ideas (Smullyan, Hodges); the proposal of mereology as a foundation of mathematics (Lesniewski, Lewis); and the mathematical analysis of quite a few foundational concepts of logic and mathematics (computability, recursiveness, consequence, probability, proof, necessity, possibility, constructiveness, programming languages, categories).
Even so, some very important questions were not solved, notably the Continuum Problem that already puzzled Cantor; the problem whether standard set theory is consistent; and the question what is the best or proper foundation of mathematics.
As to the last two questions:
There is wide agreement among mathematicians and logicians that standard set theory is consistent, and there are many proofs, for many systems X, that system X is consistent if standard set theory is (relative consistency), but there has been no proof that standard set theory is consistent, and certainly also no proof that it isn't.
And there is also wide agreement that standard set theory is sufficient to derive most of classical mathematics, and that set theory and its notation (of which there are several accepted systems) is very useful as a lingua franca for mathematics.
But one problem of standard set theory is the proliferation of an infinity of infinities that it implies (by the power set axiom and the axiom of infinity), that is both difficult to make sense of intuitively and to combine with Ockham's Razor, and apart from that there have been various other proposals of foundations for mathematics, such as Curry's formalism (and Schönfinkel and his Combinatory Logic) and Category Theory, first proposed by McLane.
It would seem that in the 21st Century, the subject that was once mathematical logic, and done on paper by a few very abstractedly minded pure mathematicians, is now mostly practised by and taught to computer scientists, next to the fact that the lingua franca of mathematics is still set theory, and effectively taught to all who learn mathematics at university level.
In any case, mathematical logic, that may be said to have started with Boole, has been shown to have amazing practical and theoretical consequences in the 20th Century, and has had huge effects on human civilization and its chances of survival (computing, internet).