Formal definition
A homomorphism between two topological groups G and H is just a continuous group homomorphism G → H. An isomorphism of topological groups is a group isomorphism which is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous. There are examples of topological groups which are isomorphic as ordinary groups but not as topological groups. Indeed, any nondiscrete topological group is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism.
Topological groups, together with their homomorphisms, form a category.

Homomorphisms
Every group can be trivially made into a topological group by considering it with the discrete topology; such groups are called discrete groups. In this sense, the theory of topological groups subsumes that of ordinary groups.
The real numbers R, together with addition as operation and its ordinary topology, form a topological group. More generally, Euclidean n-space R generated by two rotations by irrational multiples of 2π about different axes.
In every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication.

Examples
The algebraic and topological structures of a topological group interact in non-trivial ways. For example, in any topological group the identity component (i.e. the connected component containing the identity element) is a closed normal subgroup.
The inversion operation on a topological group G gives a homeomorphism from G to itself. Likewise, if a is any element of G, then left or right multiplication by a yields a homeomorphism G → G.
Every topological group can be viewed as a uniform space in two ways; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps. If G is not abelian, then these two need not coincide. The uniform structures allow one to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.
As a uniform space, every topological group is completely regular. It follows that if a topological group is T₀ (Kolmogorov) then it is already T₂ (Hausdorff), even T_3½ (Tychonoff). Many authors include the Hausdorff condition in the definition of a topological group.
Every subgroup of a topological group is itself a topological group when given the subspace topology. If H is a subgroup of G the set of left or right cosets G/H is a topological space when given the quotient topology (the finest topology on G/H which makes the natural projection q : G → G/H continuous). One can show that the quotient map q : G → G/H is always open.
If H is a normal subgroup of G, then the factor group, G/H becomes a topological group when given the quotient topology. However, if H is not closed in the topology of G, then G/H won't be T₀ even if G is. It is therefore natural to restrict oneself to the category of T₀ topological groups, and restrict the definition of normal to normal and closed.
The isomorphism theorems known from ordinary group theory are not always true in the topological setting. This is because a bijective homomorphism need not be an isomorphism of topological groups. The theorems are valid if one places certain restrictions on the maps involved. For example, the first isomorphism theorem states that if f : G → H is a homomorphism then G/ker(f) is isomorphic to im(f) if and only if the map f is open onto its image.
If H is a subgroup of G then the closure of H is also a subgroup. Likewise, if H is a normal subgroup, the closure of H is normal.
A topological group G is Hausdorff if and only if the identity subgroup is closed in G. If G is not Hausdorff then one can obtain a Hausdorff group by passing to the quotient space G/K where K is the closure of the identity. This is equivalent to taking the Kolmogorov quotient of G.
The fundamental group of a topological group is always abelian, since topological groups are examples of H-spaces, and the fundamental group of an H-space is abelian.

Relationship to other areas of mathematics

Lie group
algebraic group
topological ring