Genes and Intelligence
A few words on genomic prediction
No complex trait is 100% heritable, hence no prediction based entirely on DNA would ever be perfect. With that said, predictive genomics is progressing at a quite amazing rate right now. So while predictions can be nowhere near perfect, it is getting possible to make DNA predictions that correlate substantially with observed values.
The genes that correlate with phenotypic differences are found using genome-wide association studies (GWAS). The total effects are then aggregated up to what is called a polygenic score.
A success story --  Height
Height is highly heritable, meaning that much of the phenotypic variance (although not 100%) is due to genetic differences. Height is also highly polygenic -- it's a trait influenced by many genes, each of small effect.
Many variants that correlate with height have now been found. Polygenic scores of height correlate higher than ~0.60 with observed height. DNA predictions are generally within a few centimeters of true height (Lello, Avery, Tellier, Vazquez, de los Campos, Hsu, 2018). According to an interview with Stephen Hsu, one of the authors of this paper and co-founder of the company Genomic Prediction, they can correctly predict the height ordering of siblings within the same family 80-90% of the time (source). So based purely on DNA, can we say which of two siblings are going to be taller than the other? Not perfectly, but with decent reliably, yes.
This is about as good as predictions of complex traits can be based on current methods. There are a few principal reasons that correlations are not higher (even though it is quite impressive already in its own right). First, as I said, complex traits are not 100% heritable and therefore predictions are not expected to ever be perfect. Second, current GWAS are typically based on SNPs which are common gene variants. Rarer gene variants are expected to contribute to the variance and their effects are still needed to be uncovered. Third, only additive effects are taken into account, not gene-by-gene interactions.
Genetic IQ Prediction?
IQ, like height, is a heritable trait and is highly polygenic.
There are several approaches to finding gene variants correlated with IQ, however they can broadly be categorized into two. The most obvious approach is to simply give people intelligence tests and get their DNA. The problem with this is that it is difficult to get large samples with good intelligence tests. When this approach is used, usually a very short (say, 2 min) intelligence test is used. The second approach is to use a proxy phenotype. With this approach, the variable years of educational attainment has been used to good success, and has been shown to have a high genetic correlation with intelligence ($r_g \approx 0.7$).
Many variants that are associated with intelligence or educational attainment have been found, see e.g. (Lee et al, 2018; Savage et al, 2018). While many variants are known, IQ is not as well understood as height, and current SNP predictors correlate about ~0.3 with observed IQ (See e.g. Allegrini et al, 2018). This correlation is bound to increase in the coming few years.
Further Information
If you want a simple introduction to what can be read from the genome so far, there is a TED talk on this subject, TED2016: How to read a genome and build a human being. Although, major improvements have already been made in the years since 2016. For anyone that is more interested in the mathematical techniques used for prediction and the underlying theory, I recommend this talk by Stephen Hsu if you're further interested in genomic prediction of complex traits. I also recommend reading this review by Robert Plomin and Sophie von Stumm to get an easily understandable overview of the current state of knowledge on the subject.
References
Lello, Avery, Tellier, Vazquez, de los Campus, Hsu (2018). Accurate Genomic Prediction of Human Height. DOI: https://doi.org/10.1534/genetics.118.301267
Marty Nemko (2018). The Future of In-Vitro Fertilization and Gene Editing. Psychology Today Link.
 Lee, ..., Cesarini (2018). Gene discovery and polygenic prediction from a genome-wide association study of educational attainment in 1.1 million individuals. DOI: https://doi.org/10.1038/s41588-018-0147-3
Savage, ..., Posthuma (2018). Genome-wide association meta-analysis in 269,867 individuals identifies new genetic and functional links to intelligence. DOI: https://doi.org/10.1038/s41588-018-0152-6
Allegrini, Selzam, Rimfeld, von Stumm, Pingault, Plomin (2018). Genomic prediction of cognitive traits in childhood and adolescence. DOI: https://doi.org/10.1101/418210
Sabatini (2016). How to read a genome and build a human being. TED2016 Link to Talk. 
Hsu (2018). Genomic Prediction of Complex Traits. Youtube Link to Talk.
Plomin, von Stumm (2018). The new genetics of intelligence. DOI: https://doi.org/10.1038/nrg.2017.104