What is currently feasible in invariant theory for binary forms?
Let $F$ be a binary form of degree $d$, namely, a homogeneous polynomial of the form $$ F(\mathbb{x})=\sum_{i=0}^{d}\left(\begin{array}{c}d\\ i \end{array}\right)f_i\ x_1^{d-i}x_2^i $$ where $\mathbb{x}$ denotes the pair of variables $(x_1,x_2)$. For $g=(g_{ij})_{1\le i,j\le 2}$ in $GL_2$, define the corresponding left action on the variables by $g\mathbb{x}=(g_{11}x_1+g_{12}x_2, g_{21}x_1+g_{22}x_2)$. This gives an action on binary forms via $$ (gF)(\mathbb{x}):=F(g^{-1}\mathbb{x})\ . $$ Now consider $C(F,\mathbb{x})=C(f_0,\ldots,f_d;x_1,x_2)$ a polynomial in these $d+3$ variables. It is classically called a covariant of the (generic) binary form $F$ if it satisfies $$ C(gF,g\mathbb{x})=C(F,\mathbb{x}) $$ for all matrices $g$ in $SL_2$. Such polynomials form a ring ${\rm Cov}_d$. It has a subring ${\rm Inv}_d$ made of polynomials in the coefficients of the form $f_0,\ldots,f_d$ only. This is the ring of invariants. It is well known that invariants do not separate orbits. However, covariants separate orbits. It is also obvious that if one knows a minimal system of generators for ${\rm Cov}_d$ then it will contain (as the degree zero in $\mathbb{x}$ subset) a minimal system for ${\rm Inv}_d$.
The minimal systems for the rings ${\rm Cov}_5$ and ${\rm Cov}_6$ were determined by Gordan in his 1868 article (and not in 1875). Then von Gall determined ${\rm Cov}_8$ around 1880 and later the harder case ${\rm Cov}_7$ in 1888.
In 1967, Shioda rederived a minimal system for ${\rm Inv}_8$ and also found all the syzygies among these generators. von Gall's system for the septimic was generating but not minimal. Six elements in his list were in fact reducible. The determination of a truly minimal system of 147 covariants for ${\rm Cov}_7$ is due to Holger Cröni (2002 Ph.D. thesis) and Bedratyuk in J. symb. Comp. 2009. In 2010, Brouwer and Popoviciu obtained the minimal systems of generators for ${\rm Inv}_9$ (92 invariants) and ${\rm Inv}_{10}$ (106 invariants).
Only very recently, Lercier and Olive managed to go beyond von Gall's 1888 results and determined the minimal systems of generators for ${\rm Cov}_9$ (476 covariants) and ${\rm Cov}_{10}$ (510 covariants).
Addendum: Recently, for $d$ divisible by four, I produced an explicit list of invariants of degrees $2,3,\ldots,\frac{d}{2}+1$ and proved that they algebraically independent. See my article "An algebraic independence result related to a conjecture of Dixmier on binary form invariants".